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Astronomy & Astrophysicsmanuscript no. santiniastroph c ESO 2013November 26, 2013

The evolution of the dust and gas content in galaxies

P. Santini

1

, R. Maiolino

2,3

, B. Magnelli

4

, D. Lutz

5

, A. Lamastra

1

, G. Li Causi

1

, S. Eales

6

, P. Andreani

7,8

, S. Berta

5

,V. Buat9, A. Cooray10, G. Cresci11, E. Daddi12, D. Farrah13, A. Fontana1, A. Franceschini14 , R. Genzel5, G. Granato8,A. Grazian1, E. Le Floch12, G. Magdis15, M. Magliocchetti16 , F. Mannucci17, N. Menci1, R. Nordon18 , S. Oliver19,

P. Popesso5,20, F. Pozzi21, L. Riguccini22,23 , G. Rodighiero14, D. J. Rosario5, M. Salvato5, D. Scott24, L. Silva8,L. Tacconi5, M. Viero25, L. Wang26, S. Wuyts5, and K. Xu27

1 INAF - Osservatorio Astronomico di Roma, via di Frascati 33, 00040 Monte Porzio Catone, Italy2 Cavendish Laboratory, University of Cambridge, 19 J. J. Thomson Ave., Cambridge CB3 0HE, UK3 Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK4 Argelander Institute for Astronomy, Bonn University, Auf dem Hugel 71, D-53121 Bonn, Germany5 Max-Planck-Institut fur Extraterrestrische Physik (MPE), Postfach 1312, 85741 Garching, Germany6 School of Physics and Astronomy, CardiffUniversity, Queens Buildings, The Parade, CardiffCF24 3AA, UK7 ESO, Karl-Schwarzschild-Str. 2, D-85748 Garching, Germany8 INAF - Osservatorio Astronomico di Trieste, via Tiepolo 11, 34131 Trieste, Italy9 Aix-Marseille Universite, CNRS LAM (Laboratoire dAstrophysique de Marseille) UMR 7326, 13388 Marseille, France

10 Department of Physics & Astronomy, University of California, Irvine, CA 92697, USA11 INAF - Osservatorio Astronomico di Bologna, via Ranzani 1, 40127 Bologna, Italy12 Laboratoire AIM, CEA/DSM-CNRS-Universite Paris Diderot , IRFU/Service dAstrophysique, Bat.709, CEA-Saclay, 91191 Gif-

sur-Yvette Cedex, France13 Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA14 Dipartimento di Astronomia, Universita di Padova, vicolo Osservatorio, 3, 35122 Padova, Italy15 Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK16 INAF - IAPS, Via Fosso del Cavaliere 100, 00133 Roma, Italy17 INAF - Osservatorio Astrosico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy18 School of Physics and Astronomy, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv

69978, Israel19 Astronomy Centre, Department of Physics and Astronomy, University of Sussex, Brighton, BN1 9QH, UK20 Excellence Cluster Universe, Boltzmannstr. 2, D-85748 Garching, Germany21 Dipartimento di Astronomia, Universita di Bologna, via Ranzani 1, 40127 Bologna, Italy22 NASA Ames REserach Center, Moffett Field, CA 94035, USA23 BAER Institute, Sonoma, CA, USA24 Department of Physics & Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada25 California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA26 Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK27 NHSC, IPAC, Caltech 100-22, Pasadena, CA 91125, USA

Received .... ; accepted ....

ABSTRACT

We use deep Herschel observations taken with both PACS and SPIRE imaging cameras to estimate the dust mass of a sampleof galaxies extracted from the GOODS-S, GOODS-N and the COSMOS fields. We divide the redshiftstellar mass (Mstar)StarFormation Rate (SFR) parameter space into small bins and investigate average properties over this grid. In the first part of the workwe investigate the scaling relations between dust mass, stellar mass and SFR out to z = 2.5. No clear evolution of the dust mass withredshift is observed at a given SFRandstellar mass. We find a tight correlation between the SFR and the dust mass, which, underreasonable assumptions, is likely a consequence of the Schmidt-Kennicutt (S-K) relation. The previously observed correlation betweenthe stellar content and the dust content flattens or sometimes disappears when considering galaxies with the same SFR. Our findingsuggests that most of the correlation between dust mass and stellar mass obtained by previous studies is likely a consequence of thecorrelation between the dust mass and the SFR combined with the Main Sequence, i.e., the tight relation observed between the stellarmass and the SFR and followed by the majority of star-forming galaxies. We then investigate the gas content as inferred from dust massmeasurements. We convert the dust mass into gas mass by assuming that the dust-to-gas ratio scales linearly with the gas metallicity(as supported by many observations). For normal star-forming galaxies (on the Main Sequence) the inferred relation between the SFRand the gas mass (integrated S-K relation) broadly agrees with the results of previous studies based on CO measurements, despitethe completely different approaches. We observe that all galaxies in the sample follow, within uncertainties, the same S-K relation.However, when investigated in redshift intervals, the S-K relation shows a moderate, but significant redshift evolution. The bulk of thegalaxy population at z 2 converts gas into stars with an efficiency (star formation efficiency, SFE=SFR/Mgas, equal to the inverseof the depletion time) about 5 times higher than at z 0. However, it is not clear what fraction of such variation of the SFE isdue to an intrinsic redshift evolution and what fraction is simply a consequence of high-zgalaxies having, on average, higher SFR,combined with the super-linear slope of the S-K relation (while other studies find a linear slope). We confirm that the gas fraction(fgas = Mgas/(Mgas +Mstar)) decreases with stellar mass and increases with the SFR. We observe no evolution with redshift onceMstar andSFR are fixed. We explain these trends by introducing a universal relation between gas fraction, stellar mass and SFR thatdoes not evolve with redshift, at least out toz 2.5. Galaxies move across this relation as their gas content evolves across the cosmicepochs. We use the 3Dfundamental fgasMstarSFR relation, along with the evolution of the Main Sequence with redshift, to estimatethe evolution of the gas fraction in the average population of galaxies as a function of redshift and as a function of stellar mass: wefind that Mstar1011Mgalaxies show the strongest evolution atz 1.3 and a flatter trend at lower redshift, while fgasdecreases moreregularly over the entire redshift range probed in Mstar1011Mgalaxies, in agreement with a downsizing scenario.

Key words.galaxies: evolution, galaxies: fundamental parameters, galaxies: high-redshift, galaxies: ISM, infrared: galaxies

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1. Introduction

Dust is an important component for understanding the galaxyformation and evolution paradigm. Dust abundance is directlyconnected with galaxy growth through the formation of newstars. Indeed, dust is composed of metals produced by stellarnucleosynthesis, and then expelled into the interstellar medium(ISM) via stellar winds and supernovae explosions. A fraction

of these metals mixes with the gas phase, while about 3050%(Draine et al. 2007) of them condenses into dust grains.Therefore, dust represents a consistent fraction of the total massof metals and can be considered as a proxy for the gas metallic-ity. While dust is produced by the past star formation history, italso affects subsequent star formation, since it enhances the for-mation of molecules, hence allowing the formation of molecularclouds from which stars are produced. Moreover,dust may affectthe shape of the Initial Mass Function (IMF), through favouringthe formation of low-mass stars by fostering cloud fragmentationin low-metallicity environments and inhibiting the formation ofmassive stars (Omukai et al. 2005). Finally dust also affects thedetectability of galaxies, because it absorbs the UV starlight andreradiates it at longer wavelengths. For all these reasons, investi-gating dust propertiesand dust evolution is a powerful diagnosticto achieve a more complete view of galaxy evolution throughoutcosmic time.

With the launch of ESAs Herschel Space Observatory(Pilbratt et al. 2010), thanks to its improved sensitivity and angu-lar resolution with respect to previous instruments, it has becomepossible to investigate dust properties in large samples of galax-ies (e.g., Dunne et al. 2011;Buat et al. 2012;Magdis et al. 2012;Magnelli et al. 2013;Symeonidis et al. 2013, and many others).Its two imaging instruments, PACS (Poglitsch et al. 2010) andSPIRE (Griffin et al. 2010), accurately sample the far-infrared(FIR) and submillimetre dust peak from 70 to 500 m. In thiswork we use the data collected by two extragalactic surveys,

PEP (PACS Evolutionary Probe,Lutz et al. 2011) and HerMES(Herschel Multi-tieredExtra-galactic Survey,Oliver et al. 2012),to investigate the evolution of the dust and gas content in galax-ies from the local Universe out to z 2.5.

We first study how the dust content scales with the galaxystellar content and Star Formation Rate (SFR). Dust mass, stel-lar mass and SFR are essential parameters for understanding theevolution of galaxies. Since dust is formed in the atmosphereof evolved stars and in SN winds, we expect these parametersto be tightly linked with each other. The scaling relations be-tween dust mass, stellar mass and SFR in the local or relativelynearby (z < 0.35) Universe have been investigated by recentstudies based on Herschel data, such as Cortese et al. (2012)andBourne et al.(2012). In this work, we extend the analysis

to higher redshifts, and by enlarging the Herschel detected sam-ple by means of a stacking analysis we gain enough statisticsto study the correlations between the dust mass and either thestellar mass or the SFR, by keeping the other parameter fixedwithin reasonably small intervals. For the first time we inves-tigate the dust scaling relations by disentangling the effects ofstellar mass and those of the SFR. This resolves degeneraciesassociated with the so-called star formation Main Sequence (MShereafter). The latter is a tight correlation observed between theSFR and the stellar mass from the local Universe out to at least

Send offprint requests to: P. Santini, e-mail:[email protected]

Herschel is an ESA space observatory with science instruments pro-vided by European-led Principal Investigator consortia and with impor-tant participation from NASA.

z 3, with a roughly 0.3 dex scatter (e.g., Brinchmann et al.2004;Noeske et al. 2007;Elbaz et al. 2007;Santini et al. 2009;Karim et al. 2011;Rodighiero et al. 2011;Whitaker et al. 2012,and references therein). Galaxies on the MS are thought toform stars through secular processes by gas accretion from theIntegalactic Medium. Outliers above the MS are defined as star-bursts (e.g.,Rodighiero et al. 2011). Star formation episodes inthese galaxies are violent and rapid, likely driven by mergers

(e.g.,Elbaz et al. 2011;Wuyts et al. 2011b;Nordon et al. 2012).Despite the much more vigorous star formation activity observedin starbursts, according to recent studies (e.g.,Rodighiero et al.2011;Sargent et al. 2012;Lamastra et al. 2013a), these galax-ies play a minor role in the global star formation history of theUniverse, accounting for only 10% of the cosmic SFR densityatz 2. Since at any redshift most of the galaxies are locatedon the MS, most studies cannot investigate the dependence ofphysical quantities (e.g., dust content) on stellar mass and SFRindependently, since these two quantities are degenerate alongthe MS. To disentangle the intrinsic dependence on each of thesequantities large samples of objects are required to properly inves-tigate the dependence on SFR at any fixed Mstarand, viceversa,

the dependence on Mstarat a fixed SFR.Knowledge of the dust content can be further exploited toobtain information on the gas content, if the dust-to-gas ratiois known. In the past, most studies on the gas content in high-zgalaxies have been based on CO observations (e.g. Tacconi et al.2010, 2013; Daddi et al. 2010; Genzel et al. 2010). These studieshave allowed the investigation of the relation between the molec-ular gas mass and the SFR, i.e., the Schmidt-Kennicutt relation(Schmidt 1959;Kennicutt 1998, S-K hereafter), at different cos-mic epochs. However, these observations are time consumingand affected by uncertainties associated with the CO-to-H2con-version factor, which is poorly constrained for starburst or metal-poor galaxies (seeBolatto et al. 2013, for a review).

An alternative method to derive the gas content is to ex-

ploit the dust masses inferred from FIR-submm measurementsand convert them into gas masses by assuming a dust-to-gas ra-tio (e.g., Eales et al. 2010;Leroy et al. 2011;Magdis et al. 2011;Scoville 2012). We adopt this approach in the second part of thiswork. We convert the dust mass into gas mass by assuming thatthe dust-to-gas ratio scales linearly with the gas metallicity andthat dust properties are similar to those in the local Universe,where the method is calibrated. We estimate the gas metallic-ity from our data by exploiting the Fundamental MetallicityRelation (FMR hereafter) fitted by Mannucci et al. (2010) onlocal galaxies and shown to hold out to z 2.5. Accordingto the FMR, the gas metallicity only depends on the SFR andthe stellar mass, and does not evolve with redshift (see also

Lara-Lopez et al. 2010). With these assumptions, which will bediscussed in the text, we study the relation between the SFR andthe gas mass and investigate the evolution of the gas fractionout to z 2.5 independently of CO measurements. We note,however, that the two methods for measuring the gas mass (thedust-method and CO observations) are cross-calibrated witheach other.

A similar approach was adopted byMagdis et al.(2012) byusing Herschel data from the GOODS-Herschel survey. We im-prove over their work by also using the data in the COSMOSfield that, thanks to the large number of objects, allows us togreatly expand the stacking technique to a range of galaxy phys-ical parameters not explored byMagdis et al. (2012), and to sig-nificantly shrink the uncertainties. Moreover,while Magdis et al.

(2012) bin the data in terms of their distance from the MS at anyredshift, we bin our data in stellar mass, SFR and redshift, to

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P. Santini et al.: The evolution of the dust and gas content in galaxies

avoid the inclusion of any a-priori relation between stellar massand SFR and to study the existing trend as a function of physicalparameters.

The paper is organized as follows. After presenting the dataset (Sect.2) and the method used to compute SFR, stellar, dustand gas masses, and gas metallicities (Sect.3), we present thedust scaling relations in Sect.4, and the study of the evolution ofthe gas content in Sect.5.Finally, we summarize our results inSect.6.

In the following, we adopt the -CDM concordance cosmo-logical model (H0 = 70 km/s/Mpc, M=0.3 and =0.7) anda Salpeter IMF.

2. The data set

For this work we take advantage of the wide photometric cover-age available in three extragalactic fields: the two deep GOODSfields (GOODS-S and GOODS-N, 17 11 each) and themuch larger but shallower COSMOS field (8585). Dealingwith these fields together represents an excellent combination of

having good statistics on both bright and faint sources from lowto high redshift.Most important for the aim of this work, i.e., essential to

derive dust masses, are the FIR observations carried out byHerschel with the shorter wavelength (70, 100 and 160 m)PACS camera and the longer wavelength (250, 350, 500 m)SPIRE camera. As anticipated in Sect. 1, we use the datacollected by the two extragalactic surveys PEP and HerMES.Catalogue extraction on Herschel maps is based on a PSF fittinganalysis that makes use of prior knowledge of MIPS 24 m po-sitions and fluxes. PACS catalogues are described inLutz et al.(2011) (and references therein) and Berta et al. (2011), whileSPIRE catalogues are presented inRoseboom et al.(2010) andare updated following Roseboom et al.(2012). The 3limits1 at

100, 160, 250, 350 and 500m are 1.2, 2.4, 7.8, 9.5, 12.1 mJy inGOODS-S, 3.0, 5.7, 9.2, 12.0, 12.1 mJy in GOODS-N and 5.0,10.2, 8.1, 10.7, 15.4 mJy in COSMOS, respectively. The onlyfield which was observed at 70 m is GOODS-S. After testingthat the use of 70m photometry does not introduce any signifi-cant difference in the dust mass estimates, we ignored this bandfor consistency with the other fields.

In order to infer redshifts and other properties needed forthis study, we complement Herschel observations with pub-lic multiwavelength photometric catalogues. For GOODS-Swe use the updated GOODS-MUSIC catalogue (Santini et al.2009; Grazian et al. 2006). For GOODS-N we use the cata-logue compiled by the PEP Team and described inBerta et al.

(2010) andBerta et al. (2011), publicly available at 2

. For theCOSMOS field we use the multiwavelength catalogue presentedin Ilbert et al. (2009) and McCracken et al. (2010) and avail-able at 3. COSMOS data reduction is described inCapak et al.(2007), although the new catalogue uses better algorithms forsource detection and photometry measurements. This catalogueis supplemented with IRAC photometry fromSanders et al.(2007) and Ilbert et al. (2009) and 24 m photometry fromLe Floch et al. (2009).

All the catalogues are supplemented with either spectro-scopic or photometric redshifts. Spectroscopic redshifts are

1 In deep 160, 250, 350 and 500 m observations, rms values includeconfusion noise.

2 http://www.mpe.mpg.de/ir/Research/PEP/GOODSN multiwave3 http://irsa.ipac.caltech.edu/data/COSMOS/tables/photometry/

available for 30%, 27% and 3% of the final sam-ple, respectively, in GOODS-S, GOODS-N and COSMOS. Forthe remaining sources, we adopt the photometric redshift esti-mates publicly released with the two GOODS catalogues andthose computed by the authors for COSMOS and presented inBerta et al. (2011). The latter were computed for all sourcesrather than for the I-selected subsample released byIlbert et al.(2009), and show similar quality for the objects in common.

Photometric redshifts in GOODS-S are estimated by fittingthe multiwavelength photometry to the PEGASE 2.0 templates(Fioc & Rocca-Volmerange 1997), as presented inGrazian et al.2006and updated as inSantini et al. 2009. For GOODS-N andCOSMOS, the EAZY code(Brammer et al. 2008) was adopted,as discussed inBerta et al.(2011). We refer to the papers citedabove, as well as to Santini et al. (2012b) for more detailed infor-mation about spectroscopic and photometric redshifts and theiraccuracy.

2.1. Sample selection

In order to achieve a reliable estimate of the main physical pa-

rameters required for this analysis, we need to apply some selec-tions to the galaxy sample in the three fields.We firstly require the signal-to-noise ratio in Kband to be

larger than 10. This selection ensures clean photometry and reli-able stellar mass estimates for all sources.

Secondly, in order to estimate the SFR from an IR tracer,independent of uncertain corrections for dust extinction, we re-quirea24m detection for all galaxies (see Sect. 3.2). This is thetightest selection criterion and limits the final sample to galax-ies with relatively high star formation (3252% of the sample,depending on the field). However, although it reduces the dy-namical range probed, a SFR cut is not an issue for most of thisstudy, since we analyse trends as a function of SFR or at fixedSFR. In the latter case, the use of narrow SFR intervals prevents

strong incompleteness effects within each individual bin.Finally, we remove all known AGNs from the catalogues

( 2.5% of the total final sample), by considering X-ray de-tected sources (the AGN sample ofSantini et al. 2012b), highlyobscured AGNs detected through their mid-IR excess (follow-ingFiore et al. 2008), and IRAC selected AGNs (Donley et al.2012). Indeed, besides the cold dust heated by star formationregions, these sources host a warm dust component, which isheated by nuclear accretion processes and which might bias thedust mass estimates.

3. Parameters determination

We describe in this section how the basic ingredients of our anal-ysis, i.e., stellar masses (Mstar), SFR, dust masses (Mdust), gasmasses (Mgas) and gas metallicities, are obtained.

3.1. Stellar masses

Stellar masses are estimated by fitting observed near-UV tonear-IR photometry with a library of stellar synthetic templates(e.g. Fontana et al. 2006). We adopt the same procedure de-scribed inSantini et al. (2009): we perform a 2 minimizationofBruzual & Charlot(2003) synthetic models, parameterizingthe star formation histories as exponentially declining laws oftimescale and assuming a Salpeter4 IMF. Age, gas metallic-ity, and reddening are set as free parameters, and we use a

4 Conversion factors to a Chabrier IMF are given in Sect.3.6.

3

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Calzetti et al.(2000) or SMC extinction curve (whichever pro-vides the best fit). We refer to Santini et al. (2009) and referencestherein for more details on the stellar template library. In the fit-ting procedure, each band is weighted with the inverse of thephotometric uncertainty. SinceBruzual & Charlot(2003) mod-els do not include emission from dust reprocessing, we fit theobserved flux densities out to 5.5 m rest-frame. The redshift isfixed to the photometric or spectroscopic one, where available.

To ensure reliable stellar mass estimates, in the following weremove all sources with a reduced 2 larger than 10 (413%of the final sample, depending on the field).

Our sample spans a large redshift interval, hence the rangeof rest-frame wavelengths used to measure stellar masses isnot the same for all sources. More specifically, high-zgalaxieslack constraints at the longest rest-frame wavelengths. However,Fontana et al. (2006) have shown that the lack of IRAC bandswhen estimating the stellar mass from multi-wavelength fitting,while producing some scatter, does not introduce any systemat-ics (see alsoMitchell et al. 2013). In any case, the rest-frame Kband, essential for a reliable stellar mass estimate, is sampledeven at the highest redshifts probed by our analysis.

3.2. SFR

Star formation rates are estimated from the total IR luminosityintegrated between 8 and 1000m (LIR) and taking into accountthe contribution from unobscured SF. We use the calibrationsadopted bySantini et al.(2009) (see references therein):

SFR[M/yr] = 1.8 1010

Lbol[L]; (1)Lbol = 2.2 LUV+ LIR.

Here LUV =1.5 L(2700) is the rest-frame UV luminosityderived from the SED fitting and uncorrected for extinction.

Since Herschel detections are only available for 1125%(depending on the field) of the sample5, in order to have a con-sistent SFR estimate for a larger number of sources, we esti-mate LIR from the 24 m MIPS band (reaching 3flux limitsof 20 and 60 Jy in the GOODS fields and in COSMOS, re-spectively). Most importantly, this approach also avoids any de-generacy with the dust mass estimates, derived from Herscheldata. We fit 24 m flux densities to the MS IR template de-rived byElbaz et al. (2011) on the basis of Herschel observa-tions. This template, thanks to an updated treatment of the MIR-to-FIR emission, overcomes previous issues related with the24 m overestimate of LIR and provides a reliable estimate ofthe SFR for all galaxies (see Fig. 23 ofElbaz et al. 2011). Asa further confirmation, in AppendixBwe compare the 24 m-

based SFR with that derived by fitting the full FIR photometryand find very good agreement. This test proves that the adoptionof 24 m-based SFR does not introduce relevant biases in theanalysis. Most importantly, it provides a SFR estimate that is in-dependent of the dust and gas mass measurement and thereforeallows us to confidently investigate correlations among thesequantities.

3.3. Stacking procedure

Dust masses are computed by means of Herschel observations.Only a small fraction of the sources are individually detected

5 The statistics given in this section refers to the sample in the redshiftand stellar mass range of interest and in the area over which the analysisis carried out (see Sect.3.3).

by Herschel, and only less than 10%6 fulfill the requirements ofgood FIR sampling adopted for the dust mass estimate (see Sect.3.5). Therefore, a stacking procedure to estimate the average fluxof a group of sources is needed to perform an analysis, whichis unbiased towards the brightest IR galaxies. We describe herehow average fluxes for subsamples of sources are estimated. Inthe next section we explain how such subsamples are compiled.

The stacking procedure adopted in this work is similar to that

described by Santini et al. (2012b)andalsousedin Rosario et al.(2012) andShao et al. (2010). First of all, in each Herschel bandwe restrict to the area where the coverage (i.e., integration time)is larger than half its value at the centre of the image. This re-moves the image boundaries where stacking may be less reliabledue to the larger noise level. For each zMstarSFR bin contain-ing at least 10 sources and for each Herschel band, we stack7 onthe residual image (i.e., map from which all 3detected sourceshave been subtracted) at the positions of undetected sources (byundetected we mean below 3confidence level). Each stampis weighted with the inverse of the square of the error map. Thephotometry on the stacked PACS images is measured by fittingthe PSF, while for SPIRE images we read the value of the central

pixel (SPIRE maps are calibrated in Jy/beam), which was sug-gested byBethermin et al. (2012) to be more reliable in the caseof clustered sources. Uncertainties in the stacked flux densitiesare computed by means of a bootstrap procedure. The final av-erage flux density S is obtained by combining the stacked flux(Sstacked) with the individually detected fluxes (Si) in the samebin:

S =Sstacked Nstacked+

Ndeti=1 Si

Ntot, (2)

where Nstacked, Ndetand Ntotare the number of undetected, de-tected and total sources, respectively, in the bin.

The stacking procedure implicitly assumes that sources in

the image are not clustered. However, in the realistic casesources can be clustered with other sources either included ornot included in the stacking sample. This effect may resultin an overestimation of the flux in blended sources (see, e.g.,Bethermin et al. 2012orMagnelli et al. 2013). Given the lack ofinformation on sources below the noise level, it is not straightfor-ward to correct for this effect. However, if we are able to recog-nize its occurrence, we can ignore the bins where the stacking isaffected by confusion. For this purpose, an ad hoc simulation hasbeen put into place by the PEP Team. We briefly recall the ba-sic steps of the simulation, and refer the reader toMagnelli et al.(2013) for a more detailed description. Synthetic SPIRE fluxeswere estimated through the MS template ofElbaz et al. (2011),

given the observed redshifts and SFRs, and simulated cataloguesand maps were produced. Whenever we stack on a group ofsources on real SPIRE maps, we also stack at the same posi-tions on the simulated maps and obtain a simulated average fluxdensity (Ssim). We compare Ssim with the mean value (Sinput)of the same flux densities contained in the simulated cata-logue (previously used to create the simulated maps). FollowingMagnelli et al.(2013), if|SinputSsim|/Sinput>0.5 we reject thecorresponding bin8. The largest blending effects are seen at low

6 These fractions refer to the sample over which the analysis is per-formed (see below and Sect.3.5).

7 We use the Bethermin et al. (2010) libraries available athttp://www.ias.u-psud.fr/irgalaxies/downloads.php.

8 We verified that the trends presented in this analysis are indepen-dent of the chosen threshold.

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flux densities and in the 500 m band, as expected. The crite-rion above implies rejection of10% of the stacked fluxes at250 m, 16% at 350 m and 33% at 500 m. We also runour analysis by including these bins, to check that their rejectiondoes not introduce any bias in our results.

3.4. ThezMstarSFR grid and combination across fields

The basis of our stacking analysis is to infer an average dustmass for sources showing similar properties. To this aim, we di-vide the redshiftstellar massSFR parameter space into smallbins, and run the stacking procedure on all galaxies belongingto each bin. The ranges covered by our grid are 0.052.5 in red-shift, 9.7512 in log Mstar[M] and -0.753 in logSFR[M/yr].The boundaries of the bins, listed in TablesA.1toA.5togetherwith the abundance of sources per bin, are chosen to provide afine sampling of the Mstar-SFR parameter space and at the sametime to have good statistics in each bin. We adopt bins of 0.25dex in Mstarand 0.2 dex in SFR at intermediate Mstarand SFRvalues, where we have the best statistics, and slightly larger binsat the boundaries. This choice strongly limits the level of incom-

pleteness within each individual bin. Incompleteness issues willsimply result into bins not populated and therefore missing fromour grid (e.g., at low Mstarand SFR as redshift increases).

To combine the different fields, we stack on them simulta-neously by weighting each stamp with the relative weight map.The total number of sources in each bin and the contribution ofeach field are reported in TablesA.1toA.5.Since the statisticsare strongly dominated by the COSMOS field, we do not expectintrinsic differences among the fields to significantly affect ourresults.

For each bin of the grid we compute the average redshift,Mstarand SFR of the galaxies belonging to it, and associate thesevalues to the bin. The standard deviations of the distribution of

these parameters within the bin provide the error bars associatedwith the average values.

3.5. Dust masses

For a population of dust grains at a given temperature and with agiven emissivity, the dust mass can be inferred from their globalthermal infrared grey-body spectrum and, in particular, by itsnormalization and associated temperature. More generally, thedust thermal emission in galaxies is composed by multiple ther-mal components. In order to account for this, we use, as a de-scription of the dust emission, the spectral energy distribution(SED) templates ofDraine & Li(2007). In doing so, we im-

plicitly assume that the dust properties and emissivities of oursources are similar to those of local galaxies, on which the tem-plates were tested (Draine et al. 2007). Such assumption is sup-ported by the lack of evolution in the extinction curves, at leastout toz 4(Gallerani et al. 2010). It is also supportedby the gasmetallicity range probed by our sample ( 8.58, see Sect.3.6)and by the recent results ofRemy-Ruyer et al.(2013), claimingthat the gas metallicity does not have strong effects on the dustemissivity index. Moreover, our sample is mostly made of MSgalaxies. TheDraine & Li(2007) model is also based on the as-sumption that dust is optically thin, plausibly applicable to oursample, which does not include very extreme sources such aslocal ULIRGs or high-zsources forming a few thousands of so-lar masses per year. However, as a sanity check, we also have

used the GRASIL model (Silva et al. 1998) which includes ex-treme optically thick young starburst components, and the final

Fig. 1. Example of the fits done to estimate the dust mass.Black symbols show stacked fluxes in the bin of the zMstarSFR grid with z = [0.6, 1), logMstar[M] = [10.75, 11) andlogSFR[M/yr] = [1.4, 1.6) The blue line shows the best-fittemplate from the library ofDraine & Li(2007). For a compar-ison, the green and red curves show the fits with the GRASILmodel and with a single-temperature modified blackbody (thelatter not fitted to the shortest wavelength flux density), respec-tively. The dust mass inferred with the three libraries is indicatedin the bottom right corner. The three libraries differ in the result-ing dust masses by a roughly constant offset, but yield the same

trends.

results are unaffected (see below). Finally, Galliano et al. (2011),by studying the Large Magellanic Cloud, found that dust massesmay be systematically understimated by 50% when computedfrom unresolved fluxes. The authors ascribe this effect to possi-ble vealing of the cold dust component by the emission of thewarmer regions. However, this effect would only introduce anoffset without modifying the main results of this analysis.

According to the Draine & Li (2007) model, the inter-stellar dust is represented as a mixture of amorphous sili-cate and graphite grains, with size distribution modeled by

Weingartner & Draine (2001) and updated as in Draine & Li(2007), mimicking different extinction curves. A fraction q PAHof the total dust mass is contributed by PAH particles (with< 1000 C atoms). Although they only provide a minor contri-bution to the total dust mass, their abundance has an importanteffect in shaping the galaxy SED at short wavelengths. The ma-jority (a fraction equal to 1 ) of dust grains are located in thediffuseISMandheatedbyadiffuse radiation field contributed bymany stars. This results in a single radiation intensity U=Umin,whereUis a dimensionless factor normalized to the local ISM.The rest of the grains are localized in photodissociation regionsclose to bright stars, and exposed to multiple and more intensestarlight intensities (Umin < U < Umax) distributed as a powerlaw (U).

Following the prescriptions ofDraine et al.(2007), we builda library of MW-like models with PAH abundances qPAHin the

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range 0.474.58%, 0.0 <

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Moreover, the detailed shape of the FMR is matter of debate(e.g.Yates et al. 2012;Andrews & Martini 2013). For these rea-sons, we also tested the robustness of our results by adoptingthe redshift-dependent mass-metallicity relations published byMaiolino et al.(2008) and verified that all our results are inde-pendent of the specific description of the gas metallicity.

As suggested by previous studies, focused either on local(e.g., Draine et al. 2007; Leroy et al. 2011; Smith et al. 2012;

Corbelli et al. 2012; Sandstrom et al. 2013), z < 0.5 (e.g.,James et al. 2002) or high-z galaxies (e.g., Zafar & Watson2013; Chen et al. 2013, Cresci et al. in prep.), we considerthat a fixed fraction of metals are incorporated in dust. Withinthe metallicity range probed by our sample, this is true within0.3 dex at most. Following the parameterization provided byDraine et al. (2007), we assume that the dust-to-gas ratio (DGR)scales linearly with the oxygen abundance through the constantfactorkDGR:

DGR =kDGR (O/H)= 0.01 (O/H)/(O/H)MW==0.01 10ZZ , (3)

whereZ=12+ log(O/H) is the gas metallicity and Z =8.69 isthe Solar value (Allende Prieto et al. 2001;Asplund et al. 2009).We find almost identical results from our analysis if we applythe linear relation between log DGRand gas metallicity inferredbyLeroy et al. (2011).

The universality of the depletion factor of metals intodust is outlined by the recent work ofZafar & Watson(2013).According to their analysis, the dust-to-metal ratio can be con-sidered universal, independent of either column density, galaxytype or age, redshift and metallicity. However, De Cia et al.(2013) claim that the dust-to-metal ratio is significantly reducedwith decreasing gas metallicity at Z < 0.1Zand low columndensities. Yet, this should not be a concern for our analysis, sinceour sample does not include such low-metallicity galaxies. In a

more recent paper, Chen et al. (2013) combine constraints on thedust-to-gas ratio of lensed galaxies, GRBs and quasar absorptionsystems, and find support for a simple, linear universal relationbetween dust-to-gas ratio and metallicity.

The total gas mass (atomic +molecular, Mgashereafter) canbe computed as

Mgas =Mdust/DGR. (4)

We can finally compute the gas fraction (fgashereafter) as

fgas =Mgas/(Mgas+ Mstar). (5)

The dust content, typically negligible with respect to the gas andstellar mass components (Mdust 0.01 Mstar, see below), is ig-

nored in the computation of fgas.

4. Dust scaling relations

In this section we investigate the correlations between Mstar,SFRand Mdust, and their evolution with redshift.

4.1. Dust content vs SFR

Figure 2 shows the relation between the SFR andthe dust contentfor galaxies of different Mstarat different redshifts. A correlationbetween the dust content and the star formation activity is evi-dent at all Mstarand at all redshifts, although with some scatter,

while no clear effect is observed with the stellar mass, with binsof different Mstarsometimes overlapping (see also next section).

Fig. 2.SFR vs dust mass in different redshift ranges. Galaxiesare colour coded according to their stellar mass, as shown bythe colour bar. The dashed lines corresponds to the integratedSchmidt-Kennicutt law fitted byDaddi et al.(2010), under theassumption of Solar metallicity (see text) and converted to a

Salpeter IMF.

Before discussing the interpretation of this correlation, westress here that, not only Mdust and the SFR are estimated fromdifferent observed fluxes (Herschel and 24 m bands, respec-tively) to avoid any possible degeneracy and with intrinsicallyindependent methods, but also they are not expected to be cor-related by definition. The SFR (although in our case measuredfrom 24m observations) is in principle linked to the integratedIR luminosity, i.e., it is linked to the normalization of the far-IRspectrum. The dust mass comes from a combination of the tem-plate normalization and temperature(s), which determines the

shape; since the template library that we have used contains mul-tiple heating source components, the dust mass is not triviallyproportional to the SFR, though related to it through the dusttemperature. To verify that any observed correlation is physicaland not an obvious outcome of the relation between correlatedvariables, we run a simulation that is described in AppendixC,showing that, by starting from a completely random and uncor-related distribution of dust masses and SFRs, our method doesnot introduce any artificial correlation.

The correlation observed in Fig.2primarily tells us that thedust temperature plays a secondary role. The SFRMdust corre-lation is clearly a consequence of the S-K law, linking the SFRto the gas content. Indeed, as shown in Sect.3.6, the dust massis related to the gas mass by means of the dust-to-gas ratio. In

other words, from the S-K relation, we expect the dust mass tobe roughly proportional to the gas mass, with the gas metallicity

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Fig. 3.Dust mass vs stellar mass in different redshift ranges.Symbols are colour coded according to their SFR, as shown bythe colour bar. At each Mstar, black open circles mark the binwhich lies closest to the MS (in each Mstarinterval), and in everycase within 0.3 dex from it. The correlations between Mdustand

Mstarare rather flat when the data points are separated by meansof their SFR. The dashed lines correspond to an amount of dustequal to the maximum metal mass MZ = yZ Mstar, where

yZ 0.014, assuming the extreme case of a condensation effi-ciency of 100%, while the dotted line shows the case when only50% of the metals are depleted into dust grains.

introducing minor effects through the dust-to-gas ratio. Beforeconverting dust masses into gas masses by adopting the appro-priate dust-to-gas ratio in the next section, in order to repre-sent the S-K relation on a SFR vs Mdust plot, for the momentwe assume a constant dust-to-gas ratio for all galaxies. By us-ing equation4,the S-K law (in its integrated10 version inferred

byDaddi et al. 2010for local spirals and z 2 BzK galaxies,Daddi et al. 2004) can be written in terms of SFR as a functionof Mdustas

logSFR[M/yr]=1.31 log

Mdust[M]

DGR

+ 7.80, (6)

where the last term includes the factor (1 .8 1010) used toconvert the total infrared luminosity (the original quantity inthe expression given in Daddi et al.) into SFR, as well as theoffset of 0.15 needed to convert from a Chabrier to a SalpeterIMF (see Sect.3.6), andDGRis the dust-to-gas ratio computedfrom equation3by assuming a constant Solar metallicity. The

10 The term integrated refers to the measured power law relationbetween the gas mass and the SFR (see Sect.5.1).

Fig. 4.Average dust mass values, as indicated by the colour ac-cording to the colour bar, for bins of different SFR and Mstarindifferent redshift intervals and at all redshifts (upper right panel).Dashed lines represent MS relations of star-forming galaxiesas taken from the literature; the local MS is from Peng et al.

(2010) (computed usingBrinchmann et al. 2004data), rescaledto a Salpeter IMF, while the relations at higher redshifts are fromSantini et al. (2009). Dotted lines represent the 1(=0.3 dex)scatter of the MS relation.

dashed line in Fig. 2 shows the inferred S-K relation on the SFRMdustdiagram.

Our observational points follow reasonably well the trendexpected from the S-K law, with some scatter and a systematictrend (flatter slope) at high-z. We will discuss this in Sect. 5.1,where we also account for the variation of the metallicity (hencethe variation of the dust-to-gas ratio as a function of metallicity).

4.2. Dust vs stellar mass content

We plot in Fig.3the dust mass as a function of the stellar massin bins of redshift. When the galaxies are separated according totheir SFR (coded with different colours), the correlation foundby previous authors (e.g., at low redshift byBourne et al. 2012)becomes much flatter and sometimes even disappears, hintingthat this correlation is at least partly an indirect effect driven byother phenomena. More specifically, the MdustMstarcorrelationis partly a consequence of the MdustSFR correlation, reportedin the previous section, combined with the MS, i.e., the relationbetween SFR and Mstar. When all SFR are combined together,the low mass bins are dominated by low SFR (as a consequence

of the MS), which are associated with low Mdust (because ofthe SFRMdust relation). On the other hand, high mass bins are

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dominated by high SFR and thereforeassociated with high Mdust.This results into an apparent MstarMdust correlation. To bettervisualize this effect in studies that combine together all galaxies(i.e., without binning in a grid of SFR and Mdust), in Fig.3wehave marked with black circles the bins closest to the MS (and inevery case within 0.3 dex from it). These are the bins where thebulk of the star-forming galaxy population is concentrated, and,as expected, they show a steeper MstarMdusttrend compared to

bins of constant SFR.The dashed line in Fig.3represents the expected maximum

amount of metals (MZ = yZ M totstar, where yZ 0.014 andMtotstar is the total stellar mass formed, including the final prod-ucts of stellar evolution11) produced by stars and supernovae ex-plosions, associated with the star formation required to accountfor the observed Mstar. This is also the maximum amount ofdust that can be associated with a given Mstarin a closed boxscenario and assuming a condensation efficiency in the ejectaclose to 100%. More realistically, of these metals only about3050%(Draine et al. 2007, grey dotted line in Fig.3) are ex-pected to be depleted into dust grains. These lines give the maxi-mum amount of dust expected as a function of stellar mass if the

galaxy behaves as a closed box, and metals are condensed indust grains with reasonable/highefficiency.Most of the galaxies,in particular the high mass systems, lie below the closed boxlines. This finding qualitatively agrees with the expectations oftheoretical models for the evolution of the dust content: ratherflat MdustMstar trends, i.e., decreasing dust-to-stellar mass ra-tios as the gas is consumed and transformed into stars (see, e.g.,Eales & Edmunds 1996;Calura et al. 2008;Dunne et al. 2011).Alternatively, this result might indicate that most of the dust inthese systems is lost. In support of this scenario, independentlyof the dust information, it has been acknowledged that massivegalaxies have a deficit of metals, by a factor of a few, relative towhat must have been produced in the same galaxies(Zahid et al.2012), which is ascribed to winds that have expelled metal-rich

gas out of these massive galaxies. On the contrary, hints can beseen for low Mstargalaxies (log Mstar[M] 9.75) to show a highdust mass, close to the maximum closed box limit. Recentstudies based on SPIRE data in the local and low-z(z < 0.5)Universe support this evidence: large dust-to-stellar mass ratioswere reported bySmith et al.(2012), while anti-correlations be-tween the dust-to-stellar mass ratio and stellar mass were ob-served byCortese et al.(2012) andBourne et al.(2012). Due tothe necessity of a careful check of optical counterpart associa-tions to IR galaxies with low Mstar, we do not extend this workto such low stellar masses. The dust content in low Mstargalax-ies will be investigated by means of a dedicated analysis in aforthcoming paper.

4.3. Summary view

To give a global view of these correlations, we show in Fig.4theSFRMstarplane at different redshifts, where each bin is colourcoded according to the associated dust mass. We also show MSrelations from the literature (fromPeng et al. 2010at z 0 andfromSantini et al. 2009at high-z). This representation gives aquick overview on the scaling relations existing between M star,SFR and Mdust: a weak and sometimes absent trend of MdustwithMstarand a clear correlation between Mdustand SFR.

It is also worth noting that we observe no evidence for evo-lution of Mdust across the different redshift ranges at a given

11 The fraction of stars which goes back into the ISM is 30% for aSalpeter IMF(Treu et al. 2010).

MstarandSFR; the main difference between the various redshiftpanels in Fig.4is simply that they are populated differently. Tomake this more clear, Fig. 5shows Mdustas a function of Mstar, inbins of SFR, where the colour coding identifies different redshiftbins (note that, as a by-product, Fig.5provides further evidenceof weak/absent dependence of Mduston Mstarat a fixed SFR). Ata given MstarandSFR, there is no clear evidence for evolution ofMdust with redshift within uncertainties. We note, however, that

we cannot firmly exclude a decrease in Mdust by a factor of 2from low- to high-z, though this trend is in a few cases reversed.However, observational uncertainties on our data do not allow usto claim any redshift evolution.

It is certainly true that, on average, the overall amount ofdust in galaxies at high redshift is higher, as a consequence ofthe overall higher ISM content in the bulk of high-z galaxies(see Sect. 5.5). As a matter of fact, the normalization of theMS, representing the locus where the bulk of the populationof star-forming galaxies lies, does increase with redshift (e.g.,Santini et al. 2009; Rodighiero et al. 2010; Karim et al. 2011)and, as a consequence, the dominant galaxy population movestowards larger SFR, hence being characterized by larger dust

masses (Fig.2). However, our results indicate that galaxies withthe same properties (same SFRandsame Mstar) do not show anysignificant difference in terms of dust content across the cosmicepochs, at least out toz 2.5. In other words, dust mass in galax-ies is entirely determined by the SFR and, to a lesser extent, byMstar, and it is independent of redshift within uncertainties. Putsimply, different cosmic epochs are populated by galaxies withdifferent typical SFR and Mstarvalues, and hence are character-ized by different dust masses.

At fixed SFR, a non evolving Mdust translates into a nonevolving dust temperature (Tdust). This does not contradict theresults ofMagnelli et al.(2013), presenting only a very smoothnegative evolution in the normalization of the Tdustspecific SFR

(SSFR=SFR/Mstar) relation. They also find a stronger positiveevolution in the normalization of the relation between T dustandthe distance from the MS. However, as discussed above, the nor-malization of the MS itself increases with redshift, hence differ-ent SFRMstarcombinations are probed at different epochs.

Given the lack of any significant redshift evolution in thedust mass at fixed Mstar andSFR, it is meaningful to representall redshift bins on the same SFRMstarplane (upper right panelof Fig.4)to provide an overview of the dust content over a widerrange of Mstarand SFR. Here the dust mass is computed by aver-aging the values at different redshifts. This further confirms thetrends already outlined (Mdustdepends strongly on the SFR andweakly on Mstar), over a wider dynamic range.

5. The evolution of the gas content in galaxies

We investigate here the relation between the gas content and theSFR, as well as the evolution of the gas fraction, with the aimof understanding the processes driving the conversion of gasinto stars in galaxies throughout the cosmic epochs. We recallthat gas masses are inferred from dust mass measurements byassuming that the dust-to-gas ratio scales with the gas metal-licity, and by computing the latter by means of the FMR ofMannucci et al. (2010) (see equations34). We verified that allthe results presented below are almost unchanged if the redshift-

dependent mass-metallicity relation ofMaiolino et al. (2008) isused instead of the FMR.

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Fig. 5.Dust mass vs stellar mass in panels of different SFR. The symbol colour indicates the mean redshift of each bin, as coded bythe colour bar. No evolution with redshift is observed within uncertainties at a given Mstarand SFR.

5.1. The star formation law

We plot in Fig.6 the values of SFR as a function of gas mass.The colour code identifies bins of different redshift. For the sakeof clarity, the data points at the different redshifts are also plot-ted on separate panels on the right side. This figure is analogousto Fig.2, except that Mgas, plotted here instead of Mdust, takesinto account the dependence of the gas metallicity with stellarmass and SFR (see Sect.3.6). This, however, introduces only aminor effect (the gas metallicity changes less than a factor of 23 or less, while the dust mass spans 23 orders of magnitude).The relation shown in Fig.6can be referred to as the integratedS-K law, meaning that gas masses and SFRs are investigated val-ues rather than their surface densities, as in the original S-K law,where the SFR surface density is related to the gas surface den-sity by a power law relation. We fit the data points with the rela-

tionlogSFR = a(log Mgas 10) + b. (7)

A standard 2 fit cannot be performed on our data given theasymmetric error bars. Therefore, all over our work, we apply amaximum likelihood analysis by assuming rescaled log-normalshapes for the probability distribution functions of the variableswith the largest uncertainties (logMgasin this case) and by ignor-ing the uncertainties on the other variables. By fitting the totalsample we obtaina = 1.50+0.12

0.10and b = 1.82+0.210.20, where a boot-

strap is performed to compute the parameter 1errors. The best-fit relation is represented by the black solid line in the left panelof Fig.6.However, due to inhomogeneous sampling in SFR at

different redshifts, the fit might suffer from biases in case thereis an evolution in the slope or normalization of the relation. To

investigate such effects, we also separately fit the points in eachindividual redshift bin (coloured solid lines in the right panels of

Fig. 6). The inferred slopes monotonically decrease with redshiftfrom 1.45+0.370.41in the local Universe to 0 .76

+0.110.13at z 2, while

the normalizations increase from 1.55+0.430.47to 2.10

+0.480.52. The best-

fit parameters are given in the bottom right corners of each panelof Fig.6.

By following the theoretical model of Dave et al. (2011,2012) and the observational results ofTacconi et al.(2013), wealso attempt to fit our data points with a relation that has a sin-gle redshift-independent slope and normalization slowly evolv-ing with redshift, i.e., yielding a cosmological scaling of the de-pletion time (=Mgas/SFR):

logSFR = m(log Mgas 10) + n log(1 +z) + q. (8)

The best-fit parameters are m = 1.01+0.140.17, n = 1.40+0.850.74 andq =1.28+0.14

0.17. The dashed-triple dotted lines in the right panelsof Fig.6show the inferred relation at the median redshift in eachbin. This function provides a worse fit to the data in terms ofprobability of the solution as computed from the likelihood, withrespect to equation7.

In both cases, the evolution of the relation with redshift maybe partly caused by mixing different stellar masses, whose con-tribution strongly depends on the SFR and redshift because ofthe evolution of the MS relation.

5.1.1. Comparison with previous works

The inferred relations agree, on average, well with those fittedby previous work based on CO measurements for normal star-

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Fig. 6. Left panel: Relation between SFR and gas mass. The colour code indicates different redshift intervals, as shown by thelegend in the upper left corner. The black boxes mark bins that lie in the starburst region according to Rodighiero et al.(2011). Thesolid thick black line is the power law fit to all data, and the best-fit relation is reported in the lower right corner. The dashed anddotted grey lines show the integrated Schmidt-Kennicutt relation fitted byDaddi et al. (2010) andGenzel et al.(2010), respectively,on normal star-forming galaxies (lower curves) and on local ULIRGs and z 2 SMGs (upper curves). Curves from the literatureare converted to a Salpeter IMF. Magenta dashed-dotted lines indicate constant star formation efficiencies (i.e., constant depletiontimes) of 1 (lower curve) and 10 (upper curve) Gyr1. Right panels:Relation between SFR and gas mass in different redshift bins,indicated in the upper left corner of each panel. Symbol styles and colours are as in the left panel. The coloured solid curves arethe power law fits to the data, and the numbers in the lower right corner indicate the best-fit slope (upper) and intersection atlogMgas[M] = 10 (lower) (see equation7). The dashed-triple dotted lines show the best-fit relation given in equation8calculatedat the median redshift in each panel.

forming galaxies (Daddi et al. 2010, lower dashed grey line inFig.6,see also equation 6,andGenzel et al. (2010)12, lower dot-ted grey line), although we fit a steeper slope on all data points.The values of the best-fit slopes are independent from the galaxypopulation (i.e., consistent fits are found if starburst galaxies areremoved, see below), and of the recipe adopted for the gas metal-licity (i.e., consistent results are obtained if we assume no depen-dence on the SFR and redshift evolution of the mass-metallicityrelation). Anyhow, the broad agreement with previous studiesfor the majority of galaxies (see below) and the small disper-sion (the average absolute residual is 0.15 dex in terms of log

Mgas) shown by our data points are impressive, especially giventhe completely different and independent approaches used to de-rive the star formation law. This confirms the reliability of ourapproach of deriving gas mass estimates from dust mass mea-surements.

We remind the reader that the dust method is supposed totrace both the molecular and atomic gas (the dust-to-gas con-version factor adopted refers to the total gas mass).Bigiel et al.(2008) have measured steeper slopes for the star formation lawsin local galaxies when both the molecular and atomic gas com-ponents are considered. This may explain our steeper slopescompared to previous CO-based studies (e.g., Genzel et al.

12 We used the best-fit relation between FIR and CO luminosities intheir figure 2 and the conversions given in their table 1 to convert toSFR and Mgas, respectively.

2010;Tacconi et al. 2013). However, the fair agreement with theDaddi et al. (2010) relation (inferred from CO observations, aproxy for molecular hydrogen) is suggesting that, if the latter iscorrect, the bulk of the gas in these galaxies is in the molecu-lar phase, which is reasonable given that most of these galaxiesare vigorously forming stars and will have high pressure ISMconditions (see alsoLeroy et al. 2009;Magdis et al. 2012). Thesteeper slopes found at low redshift may be determined by alarger atomic-to-molecular gas ratio at low than at high-z(seebelow). Another possibility to explain this is the trend for the MStemplate ofElbaz et al.(2011) to slightly underpredict the SFR

in the absence of Herschel data for bright galaxies at high-z (SFR>100 M/yr, see Fig.B.1andBerta et al. 2013); by moving thedata points with the largest SFR towards lower SFR values, thiseffect might be responsible for the shallower slope measured athigh redshift. However, as it can be seen in Fig.B.1, this effectis not larger than 0.10.2 dex, and is therefore unlikely to affectour other results (on the other side, the fitted slope of the S-Klaw may be sensitive to small offsets in the SFR). As a matter offact, the results presented in this paper are very similar if otherIR templates (e.g.,Dale & Helou 2002) are used to measure theSFR from 24m fluxes or from all Herschel bands. Finally, steepslopes for the global star formation law may be explained by theresults ofSaintonge et al.(2013), who claim that the gas-to-dustratio may be 1.7 times larger atz > 2 than observed locally. This,

however, would only marginally affect our highest redshift bins,whose mean redshift value is around 2.

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Note that the fact that the slope of the global S-K rela-tion, as well as those at z < 0.6, are steeper than unity im-plies that galaxies with high star formation rates have higherstar formation efficiency (defined as SFE=SFR/Mgas, equal tothe inverse of the depletion time), even if they are regular, MSgalaxies. The magenta dash-dotted lines in Fig.6trace the lociwith SFE = 1 Gyr1 (lower line) and SFE = 10 Gyr1 (up-per line). As a consequence of the super-linear slope of the S-

K relation, moderate star-forming galaxies (SFR 1 M/yr)have a SFE approaching 1 Gyr1, while strongly star-forminggalaxies (SFR several 100 M/yr) have a SFE approach-ing 10 Gyr1, implying gas depletion timescale of a few times100 Myr. However, the SFE is more properly defined as the ra-tio of the SFR over the molecular gas content. Therefore, an-other possibility to interpret our result is that the SFR/Mmolgasstays the same, and the atomic gas content decreases in stronglystar-forming galaxies, or, in other words, the latter have a largermolecular to atomic fraction. This would be confirmed by theresults ofBauermeister et al. (2010), who observe little evolu-tion in the cosmic H I density, while the molecular componentis expected to positively evolve out to the peak of cosmic star

formation (z 23,Obreschkow & Rawlings 2009;Lagos et al.2011;Popping et al. 2013).

5.1.2. The star formation law for starburst galaxies

Symbols marked with a black box in Fig.6correspond to binswhich lie in the starburst region of the SFR vs Mstardiagram ac-cording to the selection ofRodighiero et al. (2011). They selectstarburst galaxies as sources deviating from a Gaussian logarith-mic distribution of the SSFR, having SSFR four times higherthan the peak of the distribution (associated to MS galaxies).Given the average scatter of 0.3 dex of the MS (Noeske et al.2007), these galaxies are located >2 above the MS (see Fig. 4).The effectiveness of this SSFR criterion in selecting starburst

galaxies is confirmed by semi-analytical models where starburstevents are triggered by galaxy interactions during their merg-ing histories (Lamastra et al. 2013a). Galaxies from our sam-ple located in the starburst regions do seem to follow the samestar formation law as all other galaxies. We note that the se-lection of starburst galaxies above is based on the knowledgeof the MS from the literature, rather than computed directly onthe present sample. However, this does not affect our conclu-sions. Indeed, the observed correlation between SFR and Mgasistight enough that, even in case of small variations in the lo-cation of the MS, sources selected as starburst would still fol-low the same relation (for example, results are unchanged ifthe MS fromWhitaker et al. 2012is used, despite its shallower

slope). Unless indicative of a larger fraction of atomic gas instarbursts, this result is in contrast with what suggested by pre-vious studies, mostly based on CO emission (e.g., Daddi et al.2010; Genzel et al. 2010; Saintonge et al. 2012; Magdis et al.2012;Sargent et al. 2013). The latter studies find a normaliza-tion of the star formation law 10 times higher for starburstgalaxies, implying a larger SFE. In any case, since the slope ofthe relation that we infer is larger than unity (except at z > 0.6),our result does not imply a low efficiency in converting gas intostars for galaxies located in the starburst region (see next sec-tion): starburst galaxies do have, on average, larger star forma-tion efficiency (i.e., shorter depletion times) than the bulk of star-forming galaxies (typically at lower SFR).

We note that our work does not sample the most extreme ob-

jects lying at the bright tail of the SFR distribution (all but oneof the bins selected as starbursts are located between 2and

Fig. 7.Redshift evolution of the star formation efficiency (SFE,or inverse of the depletion time). Different colours refer to dif-ferent SFRs, as shown by the colour bar. Black boxes are as inFig.6.

3above the MS). Physical properties of very extreme sources,such as local ULIRGs or high-zSMGs, are not always compli-ant with local-based expectations (see, e.g.,Santini et al. 2010)and need to be treated with ad-hoc techniques (for example,

Magdis et al. 2012claim the need of submm data to reliablyestimate dust masses of SMGs). Moreover, larger statistics isneeded. We will therefore study such extreme sources in a fu-ture work.

5.2. The evolution of the star formation efficiency

The slope of the integrated S-K relation inferred from our data isgenerally steeper than unity (except possibly at high redshift).As a consequence, the SFE for high redshift galaxies, whichare also on average more star-forming, is higher than for localgalaxies, or, equivalently, the depletion time is shorter (we hereassume negligible atomic fraction for all galaxies, but see com-

ment in Sect.5.1.1). This is illustrated in Fig.7,where the SFEis plotted as a function of redshift, and where an increase in theSFE with redshift is indeed observed, although with large scat-ter. Due to degeneracy between SFR evolution and redshift itis not clear whether the increase in the SFE with redshift trulyreflects a cosmic evolution of the SFE, i.e., galaxies of a givenSFR convert their gas into stars more efficiently at high-z, or it issimply a by-product of the slope of the S-K relation convolvedwith the higher SFR characterizing high-zgalaxies (higher nor-malization of the MS). In Fig.7galaxies with different SFRs areplotted with different colours, in an attempt to break the degener-acy between redshift and SFR. Galaxies with similar SFR showno clear internal evolution with redshift. However, due to obser-vational biases (difficulties in observingfaint sources at high-z as

well as paucity of rare bright sources in small volumes at low-z)the redshift spanned by each of these sets of points is very nar-

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row, and the dispersion very high, hence we cannot rule out areal, intrinsic evolution of the SFE in galaxies (at a given SFR).

In any case, regardless of whether the evolution of the SFE isan intrinsic redshift evolution or driven by the slope of the S-Krelation and the evolution of the SFR, the net result is that thebulk of the galaxy population (i.e., galaxies on the MS) at highredshift(z 2) do formstars with a SFE higherby a factor of5than the bulk of the population of local star-forming galaxies.This evolution is roughly consistent with the evolution of thedust mass-weighted luminosity (LIR/Mdust, proportional to theSFE except for a metallicity correction) found byMagdis et al.(2012) (a factor of 4 fromz 0 to z 2) and only slightlysteeper than the evolution of the depletion time (a factor of 3in the same redshift range) observed by Tacconi et al. (2013),likely due to the steeper S-K law inferred by us compared totheir work.

5.3. The evolution of the gas fraction

Fig.8 shows the gas fraction as a function of the stellar mass,colour coded according to the redshift, in panels of different

SFR. The gas fraction decreases with the stellar mass, as ex-pected by the gas conversion into stars in a closed-box model,and increases with the SFR, as a consequence of the S-K rela-tion (see also the results ofMagdis et al. 2012and those of thePHIBSS survey presented inTacconi et al. 2013). Most interest-ing is the lack of evolution of the gas fraction with redshift, oncegalaxies are separated according to their Mstarand SFR values.Given the assumptions made to compute the gas mass, hence gasfractions, this finding is the result of the lack of (or marginal)evolution of the dust content in bins of fixed MstarandSFR (seeFig.5), combined with a minor contribution from the gas metal-licity evolution with Mstar and SFR (the FMR,Mannucci et al.2010).

From the lack of redshift evolution of the gas fraction at fixedSFRandMstar, it follows that galaxies within a given population(identified by a combinations of SFR and Mstar), convert gas atthe same rate regardless of redshift, i.e., the physics of galaxyformation is independent of redshift, at least out to the epochsprobed by our work. This is essentially a consequence of theunimodal inferred S-K relation, but Fig.8shows the result moreneatly by also slicing the relation through the dependence onstellar mass, which is the third fundamental parameter. We notethat this does not contradict the evolution of the SFE observed inFig. 7,wheredifferent stellar masses and SFR are mixed togetherand where selection effects cause the different SFR bins to bepopulated differently at different redshifts (hence the average ateach redshift is certainly biased).

In summary, our result implies that, at fixed stellar mass, theSFR is uniquely driven by the gas fraction via the star forma-tion law. In other words, if two among SFR, M star and Mgasareknown, the third property is completely determined and does notdepend on redshift. This provides a powerful tool to overcomethe observational difficulties related with the measurement of gasor dust masses and analyse the gas content for much larger sam-ples of galaxies.

5.4. Thefundamental fgasMstarSFR relation

Given the lack of evolution with redshift observed for the gasfraction once galaxies with the same Mstar andSFR are consid-

ered, we can combine all redshift bins together to increase thestatistics and infer more clearly the trend of fgas as a function

Fig. 9.Average gas fractions, as indicated by the upper colourbar, in bins of Mstarand SFR.

Fig. 10. Parameterization of the gas fraction as a function ofstellar mass at all redshifts in different SFR intervals, using thefunctional shape given in equation9(see text). Curves of differ-ent colours refer to different SFR bins, as shown by the colourbar.

of Mstarin different SFR intervals. Figure9shows the resultingglobal dependence of the gas fraction (given by the colour cod-ing) on the SFRMstarplane. In each SFR interval, we fit the datapoints with a linear relation in the logarithmic space :

logfgas = +(logMstar 11). (9)

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Fig. 8.Gas fraction vs stellar mass in panels of different average SFR. The colour of the symbols reflects the mean redshift ofeach bin, as indicated by the colour bar. No evolution with redshift is observed, within uncertainties, at given SFR and Mstar. Greydashed curves are the best-fits to the data assuming the functional shape in equation9.Best-fit parameters for each SFR interval aresummarized in Table1.

We shift the stellar masses, placing them across zero, in orderto de-correlate the slope and offset parameters in the linear fitresult. The best-fit parameters are given in Table1,and the best-fit curves are shown by the dashed grey lines in Fig. 8and alsoby the solid coloured lines in Fig.10, which provides a directcomparison at different SFRs. We note that the functional formadopted above does not necessarily have physical meaning: itis a purely phenomenological representation of the data to bettervisualize the observed trends and to interpolate the three physicalquantities for later use of this 3D relation.

The three-dimensional fgasMstarSFR relation shown inFig.10is a fundamental relationthat does not evolve with red-shift, at least out to z 2.5. Galaxies move over this surfaceduring their evolution.

Fig.11shows a 3D representation of such a relation. Furtherinvestigation of this 3D relation and its physical interpretationgoes beyond the scope of this paper, and will be discussed ina future work, as well as the relation between the independentquantities Mgas, Mstar and SFR. Here we only emphasize thatthe redshift evolution of the S-K law investigated in equation8seems to disappear once sources are divided in bins of Mstar.Indeed, the redshift evolution of the SFE illustrated in Fig. 7ismost likely a consequence of the fact that high-zbins are mostlypopulated by galaxies with high SFR, which are characterizedby high SFE, as a consequence of the super-linear slope of theS-K relation.

We note that the fundamental relation presented here is in-

deed a physical result, rather than just a way of looking at theredshift evolution through the evolution of another parameter

Table 1. Best-fit parameters of the functional shape in equation 9describing the gas fraction as a function of the stellar mass indifferent SFR intervals.

logSFR[M/yr] logMstar min

0.25 0.25 2.17+0.160.31 1.04

+0.320.37 9.85

0.25 0.50 1.53+0.330.35 0.52

+0.390.39 9.89

0.50 0.75 1.34+0.140.19 0.53

+0.200.25 9.88

0.75 1.00 1.58+0.020.02 0.85

+0.040.05 9.89

1.00 1.20 1.38+0.030.02 0.79

+0.090.10 9.90

1.20 1.40 1.34+0.050.05 0.86

+0.080.08 9.90

1.40 1.60 1.22+0.050.05 0.77

+0.100.09 10.15

1.60 1.80 1.06+0.030.03 0.79+0.050.08 10.151.80 2.00 0.96+0.02

0.02 0.76+0.110.12 10.39

2.00 2.25 0.85+0.060.05 0.82

+0.180.15 10.40

2.25 2.50 0.75+0.060.02 0.70

+0.070.18 10.40

2.50 3.00 0.54+0.050.03 0.50

+0.020.15 10.66

Notes.The last column reports the minimum stellar mass sampled ineach SFR bin. These parameterizations should not be employed belowthese limits.

(e.g., SFR). In other words, the inclusion of the SFR or stel-lar mass as parameters is not masking a true underlying redshift

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Fig. 12.Left: Gas fraction vs Mstar at different redshifts (in different colours according to the legend) for Main Sequence (MS)galaxies.Right: Gas fraction vs redshift at different Mstar(in different colours according to the legend) for MS galaxies. Curves are

obtained by interpolating the fgasparameterizations reported in Fig. 10and Table1and the MS relations (see text for details) atMstarabove the minimum sampled Mstarcommon to all SFR bins. Mean uncertainties on gas fraction associated to main sequencegalaxies in each redshift (left) or stellar mass (right) bin are plotted.

Fig. 11. Representation of the 3D fundamental fgasMstarSFRrelation. The colour code indicates the average SFR of each bin.The best-fit relations shown in Fig.8are overplotted.

evolution. As a matter of fact, no similar relation is obtained ifredshift is replaced to either SFR or Mstar.

5.5. The evolution of the gas fraction among Main Sequencegalaxies

The finding that the fundamental fgasMstarSFR relationdoesnot evolve with redshift does not contradict the claimed red-shift evolution of the gas fraction in galaxies (e.g., Daddi et al.2010;Tacconi et al. 2010,2013;Magdis et al. 2012). Indeed, asalready mentioned, galaxies do not uniformly populate this 3Dsurface. As they evolve, the bulk of star-forming galaxies pop-

ulate different regions of this surface, as a consequence of gasaccretion, gas consumption by star formation and gas ejection.The projection of such a distribution onto the MstarSFR planeyields the MS and its evolution with redshift.

As suggested by various models, the evolution of galaxies islikely driven by the evolution of their gas content. The evolutionof the MS is likely a by-product of the gas content through theS-K relation, or more generally through the fundamental fgasMstarSFR relationillustrated above. While the evolution of theMS has been constrained by several observations,its driving pro-cess, which is the evolution of the gas content, is still looselyconstrained. We can however exploit the observed evolution ofthe MS to infer the evolution of the gas fraction of the population

of galaxies dominating star formation at any epoch, by exploit-ing thefundamental fgasMstarSFR relation.We take advantage of the mathematical representation of the

gas fraction as a function of Mstarat given SFR shown in Fig.10,and we linearly interpolate these relations onto a finer SFR grid.We then adopt the MS relations reported in Fig. 4and linearlyinterpolate them onto a fine redshift grid. At a given Mstar andredshift, we use the MS relation to compute the expected SFR,according to which we select the appropriate fgasparameteriza-tion.

The resulting evolution of fgas with stellar mass at differentredshifts (colour coded) is shown in the left panel of Fig. 12.The orthogonal plot, i.e., the redshift evolution of fgasfor differ-ent stellar masses (colour coded), is shown in the right panel

of Fig. 12. These plots illustrate how the bulk of the star-forming galaxy population at various epochs populates the 3D

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fundamental fgasMstarSFR relationas a function of redshift.Essentially, for a given stellar mass, the average gas content ofstar-forminggalaxies increases steadily with redshift, at least outtoz 2.5. The increase rate is steeper for low mass galaxies withrespect to massive galaxies. Galaxies with logMstar[M] 10.6reach fgas 0.25 around the peak of cosmic star formation at

z 2.5, while massive galaxies, with log Mstar[M] 12 reacha gas fraction of only 0.15 at the same cosmic epoch. This be-

haviour is consistent with a downsizing scenario (Cowie et al.1996; Fontanot et al. 2009), where massive galaxies have al-ready consumed most of their gas at high redshift, while lessmassive galaxies have a larger fraction of gas (more complexscenarios resulting from the interplay of inflows, outflows andstar formation are not excluded). Further, in massive galaxies thegas fraction decreases more steeply, moving towards lower red-shift (with respect to low mass galaxies) and their gas evolutionflattens to low values atz 1.3. Instead, low mass galaxies showa shallower and more regular decrease of the gas content, mov-ing towards lower redshifts. Both trends are further indicationsof downsizing.

The fgas values are somewhat lower by a factor of1.52

on average (after accounting for the IMF conversion) than in-ferred by the high-zCO survey ofTacconi et al.(2013). A sim-ilar or even larger mismatch with CO-based results was foundbyConselice et al.(2013), who compute gas fractions from SFRand galaxy sizes by inverting the S-K law. We ascribe the dis-crepancy to the combination of the various uncertainties asso-ciated with CO studies and with our method. In addition, theunderestimate by 50% of the dust mass of unresolved sourcesfound byGalliano et al. (2011) may also explain the lower val-ues found by us. The gas fractions derived by us are also lowerby a factor of 2 than those published byMagdis et al.(2012),who adopt a similar method. This might be caused by cosmicvariance effects: based on the two GOODS fields only, the anal-ysis ofMagdis et al.(2012) may be affected by statistical uncer-

tainty. The inclusion of COSMOS data provides much improvedstatistics that is crucial in stacking analyses. Indeed, the stack-ing result is closely related to the number of stacked sources.Even if COSMOS is shallower than the deep GOODS fields,SPIRE observations, on which dust masses mostly rely, are con-fusion limited. Therefore, the statistics is strongly dominated byCOSMOS. To verify whether cosmic variance effects could beresponsible for such disagreement, we repeated our analysis byonly including the two GOODS fields. Given the limited statis-tics, we end up with only 10 data points. We compared thesewith our gas fractions and found that in 30% of the cases theformer are indeed larger by a factor of 22.5, while the rest ofthe points are consistent within their error bars. Finally, we note

that the disagreement with previous works is reduced when theGRASIL model is adopted instead ofDraine & Li(2007).

5.6. Comparison with theoretical predictions

The evolution of the gas fraction is a powerful observable totest the various physical processes at play in galaxies and im-plemented by theoretical models, such as star formation, gascooling and feedback. Here we compare our findings for theevolution of the gas fraction with the expectations of the semi-analytical model of galaxy formation developed byMenci et al.(2008) (and references therein). This connects, within a cosmo-logical framework, the baryonic processes (gas cooling, star for-mation, supernova feedback) to the merging histories of the dark

matter haloes, computed by means of a Monte Carlo simula-tion. Gas is converted into stars through two main channels: a

steady (or quiescent) accretion mode, in which the cold gas in thegalaxy disk is converted into stars on long timescales (1 Gyr),and an interaction-driven mode, where gas destabilized duringmajor and minor mergersand fly-by events is convertedinto starson shorter timescales (107 yr; seeLamastra et al. 2013a,bfor amore detailed description). AGN activity triggered by the samegalaxy interactions and the related feedback processes are alsoincluded.

The predicted gas fraction as a function of stellar mass andredshift is shown in Fig.13. On the same figure we report the ex-trapolations for MS galaxies based on our observations alreadyshown in Fig.12. As discussed above, MS galaxies represent thebulk of the galaxy population and can be directly compared tothe darkest contours, enclosing the region occupied by most ofthe galaxies.

Observations are generally well reproduced by the theoreti-cal model, although with some systematic deviations. The trendswith both stellar mass and redshift are recovered, as well as thedownsizing expectations: a strong evolution can be noticed inlow mass galaxies (Mstar 1011M/yr), which are gas-rich outtoz 1 (bottom right panel), while progressively more massive

galaxies have already consumed their gas at this epoch (upperright panel). While a very good agreement is recovered for allstellar masses at high redshift (z 2, upper-left panel), the pre-dicted evolution of the gas fraction is more regular than observedat intermediate redshifts, with a gas fraction in log Mstar[M]11.5 galaxies of 0.2 at z 0.6, around twice the observedvalue (central left panels). The overall systematic gas richnessof model galaxies compared to the observations relates to thelong-standing problem of theoretical models in reproducing thegalaxy stellar mass functions at high redshift. Indeed, the num-ber of massive galaxies is underpreticted by the models (e.g.Fontanot et al. 2009;Santini et al. 2012a), consistently with theinefficiency of the gas conversion and mass buildup processesin the distant Universe. Once gas consumption has started, it isnot efficiently suppressed at late stages. Indeed, the model pre-dicts a fraction of very massive (log Mstar[M] 11.5) galaxieswhich are still gas-rich at z < 1, at variance with what observed(lower- and central-left panels and top-right one). Although itcan be partly ascribed to fluctuations in the fgasdistribution gen-erated by the low number statistics of such high M stargalaxies,this behaviour is a manifestation of a known problem commonto all theoretical models, in which the suppression of the starformation activity is still inefficient, despite the feedback pro-cesses at work. This is related to the difficulties in reproducingthe fraction of red passive galaxies(Fontana et al. 2009).

For all these reasons, the comparison of observed and mod-eled gas fraction is of major importance to constrain the phys-

ical processes implemented in models of galaxy formation andevolution. A more detailed and complete comparison with theo-retical expectations will be tackled in a future work.

6. Summary

We have used Herschel data from both PACS and SPIRE imag-ing cameras to estimate the dust mass of a large sample of galax-ies extracted from the GOODS-S, GOODS-N and COSMOSfields. To explore a wide range of galaxy properties, includ-ing low mass and moderate star-forming galaxies, we have per-formed a stacking analysis on a grid of redshifts, stellar massesand SFR, and considered average values. With these outputs we

have studied the scaling relations in place between the dust con-tent of galaxies and their stellar mass and SFR at different red-

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Fig. 13. Predicted evolution of the gas fraction according to thesemi-analytical model ofMenci et al. (2008). The five filled con-tours indicate the fraction of galaxies having a given fgas at a

fixed Mstar(leftpanels) and redshift (rightpanels). The trends forMS galaxies extrapolated by our observations (shown in Fig.12)are overplotted.

shifts, from the local Universe out to z =2.5. Our main resultsare the following.

No clear evolution of the dust mass with redshift is observedat a given SFR andstellar mass. Although there is a globalredshift evolution of the dust content in galaxies, as a conse-quence of the increased ISM content at high-z, our findingsindicate that galaxies with the same properties (same SFRandsame Mstar) do not show any significant difference in

terms of dust content across the cosmic epochs, at least outto z 2.5. In other words dust mass in galaxies is mostlydetermined by SFR and Mstarand is independent of redshift.

The dust content is tightly correlated with the star formationactivity of the galaxy. This correlation is in place at all val-ues of Mstarprobed and at least out toz 2.5. Under the as-sumption that the dust content is proportional to the gas con-tent (with a factor scaling with the gas metallicity), the ob-served correlation is a natural consequence of the Schmidt-Kennicutt (S-K) law.

The correlation between the dust and stellar mass observedby previous studies (which averaged together all SFR) be-comes much flatter or even disappears when taken at a fixedSFR. The MdustMstarrelation is at least partly a result of the

MdustSFR correlation combined with the Main Sequence(MS) of star-forming galaxies.

We have then taken one step further and computed gasmetallicities from the stellar mass and the SFR accordingto the Fundamental Metallicity Relation (FMR) fitted byMannucci et al.(2010), and estimated gas masses by assumingthat the dust-to-gas ratio linearly scales with the gas metallic-ity. We note that all our results are robust against the specificparameterization chosen to describe the gas metallicity (e.g.,FMR against redshift-dependent mass-metallicity relation). This

method provides a complementary approach to investigate thegalaxy gas content independently of CO observations. Under ourassumptions we find the following.

We fit a power law relation between the SFR and the gasmass, in good agreement with that previously obtained byDaddi et al.(2010), and also broadly consistent with the re-sults ofGenzel et al. (2010). This agreement is remarkable,given the completely different approach between our studyand the two works above based on CO measurements. Wefind that all galaxies follow the same star formation law (in-tegrated S-K law), with no evidence of starbursts lying on anoffset relation, though our sample lacks the most extremely

starbursting sources (such as local ULIRGs and their ana-logues at high-z). The slope of this relation is on averagesteeper than unity, implying that strongly star-forming galax-ies have higher star formation efficiency (SFE, i.e., the in-verse of the depletion time), or shorter depletion time. Wealso find a mild, but significant evolution of the S-K law withredshift.

We observe an evolution of the SFE with redshift, by abouta factor of 10 from z 0 to z 2.5. This applies to thebulk of the galaxy population dominating star formation ateach epoch. However, it is not clear whether such e

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