Dust and Gas in Star Forming Galaxies to z = 2

# The Evolving Interstellar Medium of Star Forming Galaxies since z=2 as Probed by Their Infrared Spectral Energy Distributions

Georgios E. Magdis 11affiliation: Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH , E. Daddi 22affiliation: CEA, Laboratoire AIM, Irfu/SAp, F-91191 Gif-sur-Yvette, France , M. Béthermin 22affiliation: CEA, Laboratoire AIM, Irfu/SAp, F-91191 Gif-sur-Yvette, France , M. Sargent 22affiliation: CEA, Laboratoire AIM, Irfu/SAp, F-91191 Gif-sur-Yvette, France , D. Elbaz 22affiliation: CEA, Laboratoire AIM, Irfu/SAp, F-91191 Gif-sur-Yvette, France , M. Pannella 22affiliation: CEA, Laboratoire AIM, Irfu/SAp, F-91191 Gif-sur-Yvette, France , M. Dickinson 33affiliation: NOAO, 950 N. Cherry Avenue, Tucson, AZ 85719, USA , H. Dannerbauer 44affiliation: Universität Wien, Institut ür Astronophysik, Türkenschanzstrasse 17, 1180 Wien, Austria , E. da Cunha 55affiliation: Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany , F. Walter 55affiliation: Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany , D. Rigopoulou 11affiliation: Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH 66affiliation: Space Science & Technology Department, Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire OX11 0QX , V. Charmandaris 77affiliation: Department of Physics & ICTP, University of Crete, GR-71003, Heraklion, Greece 88affiliation: IESL/Foundation for Research & Technology-Hellas, GR-71110, Heraklion, Greece 99affiliation: Chercheur Associé, Observatoire de Paris, F-75014, Paris, France , H.S Hwang 1010affiliation: Smithsonian Astrophysical Observatory, 60 Garden Street, Cambridge, MA 02138, USA J. Kartaltepe 33affiliation: NOAO, 950 N. Cherry Avenue, Tucson, AZ 85719, USA
###### Abstract

Using data from the mid-infrared to millimeter wavelengths for individual galaxies and for stacked ensembles at , we derive robust estimates of dust masses () for main sequence (MS) galaxies, which obey a tight correlation between star formation rate (SFR) and stellar mass (), and for star-bursting galaxies that fall outside that relation. Exploiting the correlation of gas to dust mass with metallicity (/), we use our measurements to constrain the gas content, CO-to-H conversion factors () and star formation efficiencies (SFE) of these distant galaxies. Using large statistical samples, we confirm that  and SFE are an order of magnitude higher and lower, respectively, in MS galaxies at high redshifts compared to the values of local galaxies with equivalently high infrared luminosities (  L). For galaxies within the MS, we show that the variations of specific star formation rates (sSFR=SFR/) are driven by varying gas fractions. For relatively massive galaxies like those in our samples, we show that the hardness of the radiation field, , which is proportional to the dust mass weighted luminosity (/), and the primary parameter defining the shape of the IR-spectral energy distribution (SED), is equivalent to SFE/Z. For MS galaxies with stellar mass we measure this quantity, , showing that it does not depend significantly on either the stellar mass or the sSFR. This is explained as a simple consequence of the existing correlations between SFR, and SFR. Instead, we show that (or equally /) does evolve, with MS galaxies having harder radiation fields and thus warmer temperatures as redshift increases from to 2, a trend which can also be understood based on the redshift evolution of the and SFR relations. These results motivate the construction of a universal set of SED templates for MS galaxies, that are independent of their sSFR or , but which vary as a function of redshift with only one parameter, .

slugcomment: to appear in ApJ

## 1. Introduction

Deep and wide multi-wavelength extragalactic surveys have greatly enhanced our understanding of galaxy evolution. It has now been well established that the star formation rates (SFRs) in galaxies were on average higher in the past, with galaxies emitting the bulk of their bolometric energy in the infrared (IR) progressively dominating the star formation density of the Universe that peaked at (e.g., Le Borgne et al. 2009).

A recent major step forward in characterizing the nature of star formation in distant galaxies has been the discovery that the majority of star-forming galaxies at every redshift define a narrow locus in the stellar mass ( star formation rate (SFR) plane. This correlation has been observed in the local Universe (Brinchmann et al. 2004; Peng et al. 2010), as well as at intermediate redshifts (Noeske et al. 2007; Elbaz et al. 2007; Daddi et al. 2007; Pannella et al. 2009; Rodighiero et al. 2010a, Karim et al. 2011, Magdis et al. 2010), and beyond (e.g., Daddi et al. 2009, Stark et al. 2010), with a normalization factor that increases rapidly with look back time, mirroring the increase in the star formation activity of the early galaxies.

The fact that this correlation between SFR and appears to be present at all epochs has been used to define a characteristic specific star formation rate () at each redshift and stellar mass, and from its evolution with cosmic time, a main sequence (MS) mode of star formation that is followed by the majority of star-forming galaxies. This main sequence also serves as a tool to distinguish starburst (SB) from normal111Throughout the paper we will use the terms “normal” and “main sequence” galaxy interchangeably galaxies at any redshift, simply by measuring the excess sSFR of the galaxy with respect to the SFR– correlation at that redshift. Such outliers are known to exist at any redshift. In the local Universe, a class of galaxies that are usually identified as starbursts are the (Ultra) Luminous Infrared Galaxies (ULIRGs,   L, Sanders & Mirabel 1996, Elbaz et al. 2007). However, in the distant Universe it has proven to be harder than initially thought to associate pure starbursts with a specific class of high galaxies, although at least a fraction of Submillimeter Galaxies (SMGs) seem to exhibit elevated specific star formation rates (Tacconi et al. 2006; 2008; Daddi et al. 2007; 2009, Takagi et al. 2008).

The remarkable uniformity of MS galaxies, as indicated by the SFR- correlation, and their contrasting nature to that of starburst galaxies, has recently been further manifested with respect to two more sets of observable parameters. Using direct measurements of the total infrared luminosity (=)  of distant galaxies that have now become possible with the Herschel Space Observatory (Pilbratt et al. 2010), Elbaz et al. (2011) showed that galaxies that follow the SFR- correlation are also part of a secondary, infrared main sequence, defined by a universal total-to-mid-IR luminosity ratio, IR8 /, where is the rest-frame 8m luminosity. The majority of MS sequence galaxies, at all luminosities and redshifts, are found to obey this linear correlation between  and , suggesting that they share a common IR spectral energy distribution (SED), with a mid-to-far-IR shape that has evolved little with cosmic time. Interestingly, starburst galaxies, which systematically fall above the SFR- main sequence, are also outliers to the - relation, exhibiting elevated IR8 values with respect to normal galaxies.

The third quantity that highlights the distinct nature of MS and starbursts encompasses the main driver of star formation activity: the molecular gas mass (), i.e., the raw material out of which galaxies form stars. Recent studies, by Daddi et al. (2010) and Genzel et al. (2010), found that normal galaxies at any redshift have lower star formation efficiencies (/), compared to star-bursting systems that exhibit an accelerated mode of star formation activity, probably triggered by a major merger. These studies all suggest that there are two regimes of star formation: i) a long–lasting mode, followed by the majority of galaxies at every redshift that also form a tight MS in the SFR and in the planes, and ii) a short–lived starburst mode for galaxies that can become strong outliers from both relations. However, this picture heavily relies on  estimates which are still a matter of significant debate due to the poorly determined conversion factor to derive molecular gas mass from CO luminosities (= /), which is known to vary as a function of metallicity and, perhaps, of the intensity of the radiation field (e.g., Leroy et al. 2011, Magdis et al. 2011b).

Although the separation between starbursts and MS galaxies is already apparent in the observed - plane, the distinct nature of the star formation activity of the two populations becomes truly evident when different  factors are used for the two classes of galaxies. This differentiation of the  value has clearly been seen in the nearby universe, with local ULIRGs (starbursts) having, a value of  that is a factor of 6 smaller than that for local spiral galaxies222We note that Papadopoulos et al. (2012) presented evidence that higher  values are possible in local ULIRGs (e.g., Downes & Solomon 1998), and seems to be true at high redshift too. Using a dynamical and a dust-to-gas mass ratio analysis of star-forming disks at , Daddi et al. (2010a) and Magdis et al. (2011b) argued for a CO conversion factor333The units of , pc (K km s), are omitted from the text for brevity. Also,  estimates in this work account for the presence of Helium coexisting with the molecular hydrogen   4.0 for this kind of objects, similar to that of the Milky Way and local spirals. On the other hand, several studies place an upper limit of   0.8 for several SMGs (e.g., Tacconi et al. 2008, Carilli et al. 2010, Magdis et al. 2011b, Hodge et al. 2012). Furthermore, recent numerical simulations indicate a clear distinction between the  values for disks and mergers at all redshifts, although with a considerable scatter (e.g., Narayanan et al. 2011, Feldmann et al. 2012). These findings challenge the common approach of “blindly” applying a (local) ULIRG-like   value to derive the  of high ULIRGs. Instead, they highlight the necessity of determining  in larger samples of high galaxies and of investigating how it varies between normal galaxies and mergers/starburst, but also among galaxies on the main sequence.

To this end, in Magdis et al. (2011b), we applied, for the first time at high redshift, a method for measuring  that is commonly used in the local universe. The method relies on measuring the total dust mass of a galaxy () and assuming that it is proportional to  (e.g., Leroy et al. 2011). Taking advantage of the detailed characterization of the peak of the SED enabled by Herschel data, as well as of the Rayleigh-Jeans tail from ground based mm observations, we acquired robust  estimates for a normal disk at and a star-bursting SMG at . The dust mass estimates were subsequently used to infer an  value of 4.0 for the disk and an upper limit of 1 for the starburst galaxy, in agreement with previous dynamical estimates (Daddi et al 2010a, Hodge et al. 2012). This study also offered a first hint of a close link between the SED shape, as traced by the dust- ghted luminosity (/) and  and SFE, with MS galaxies exhibiting lower / values compared to SB galaxies.

Having demonstrated the applicability of the method at high redshift, here we extend this exercise to a larger sample of main sequence disks at and that have extensive photometry from rest-frame UV to mm wavelengths, including state of the art Herschel data from the Herschel Great Observatories Origins Deep Survey (GOODS-Herschel, PI D. Elbaz) as well as low-transition CO observations and mm continuum interferometric measurements. We use this to investigate the average  value for high MS galaxies, finding consistent values with what is predicted by the local  versus Z trend and what is inferred by dynamical analyses (Daddi et al. 2010b).

Pushing the method further, we here analyse large statistical samples of galaxies at and to derive their average dust and gas mass content. While the lack of CO measurements for such large samples prevent us from learning anything on their conversion factor, the recovered information is invaluable to study the distribution of gas in high redshift galaxies. This allows us to investigate possible variations of gas fractions and SFE within the MS, providing insights about the origin of the “thickness” of the SFR- relation that has been shown to reflect the variation of the physical properties of MS galaxies (e.g., Salmi et al. 2012, Elbaz et al. 2011). Furthermore, the available data allow us to investigate the SED shape of MS galaxies as function of redshift and also as a function of their their offset from the main sequence. Put together, in this study we attempt to characterize the gas, dust and SED properties of main sequence galaxies throughout cosmic time and therefore provide a better understanding of the nature of the sources that dominate the star formation density at all redshifts (e.g., Rodighiero et al. 2011, Sargent et al. 2012).

This paper is organized as follows. In Section 2, we describe our sample and the multi-wavelength observations. In Section 3 we present a detailed analysis of the far-IR properties. In Section 4 we discuss the method to derive  and present  and SFE estimates for MS and starburst galaxies. In Section 5, we explore the variations of SFE within the MS, and discuss two possible scenarios to explain the observed dispersion of the SFR- correlation. In Section 6 we investigate these two scenarios through stacking, while in Section 7 we explore the shape of the SED of MS galaxies as a function of cosmic time and build template SEDs of MS galaxies in various redshift bins. Finally, in Section 8, we provide a discussion motivated by the results of this study, while a summary of the latter is presented in Section 9. Throughout the paper we adopt km s Mpc, = 0.7 and a Chabrier IMF (Chabrier et al. 2003).

## 2. Observations and Sample Selection

The aim of this study is to investigate the far-IR and gas properties of high, normal, main sequence galaxies. Instead of using a large, heterogeneous sample, we decided to focus on a small but well defined set of sources selected to meet the following criteria: 1) available spectroscopic redshifts; 2) rich rest-frame UV to mid-IR photometry; 3) no excess in the specific star formation rate, i.e., sSFR/sSFR 3 (where sSFR is the MS trend, as detailed below) 4) available [1-0] or [2-1] low-transition CO observations, to enable CO luminosity estimates, without the caveat of the uncertainties introduced by the excitation corrections affecting higher CO transition lines; 5) Herschel detections that offer a detailed sampling of the far-IR part of the SED; and, if possible, 6) mm continuum data that provide a proper characterization of the Rayleigh-Jeans tail, which is crucial for robust dust mass estimates.

To derive a characteristic sSFR at a given redshift and a given stellar mass, we define a main sequence, SFR, that varies with stellar mass with a slope of 0.81 (e.g., Rodrighero et al. 2011), and evolves with time as (e.g., Elbaz et al. 2011, Pannella et al. 2009). In what follows we describe the Herschel observations used in this study, as well as the sample of galaxies considered here.

### 2.1. Herschel  Data

We use deep 100 and 160m PACS and 250, 350, and 500m SPIRE observations from the GOODS-Herschel program. Details about the observations are given in Elbaz et al. (2011). Herschel fluxes are derived from point-spread function (PSF) fitting using galfit (Peng et al. 2002). A very extensive set of priors, including all galaxies detected in the ultra-deep Spitzer Multiband Imaging Photometer (MIPS) 24m imaging, is used for source extraction and photometry at 100, 160 and 250m, which effectively allow us to obtain robust flux estimates for relatively isolated sources, even beyond formal confusion limits at 250m. For 350 and 500m, this approach does not allow accurate measurements due to the increasingly large PSFs. Hence, we use a reduced set of priors based primarily on Very Large Array (VLA) radio detections, resulting in flux uncertainties consistent with the confusion noise at the IR wavelengths. Our measurements are in good agreement with the alternative catalogs used in Elbaz et al. (2011). The advantage of using galfit for PSF fitting is in its detailed treatment of the covariance matrix to estimate error bars in the flux measurements, which is crucial to eventually derive reliable estimate of flux errors for the case of blended/neighbouring sources. The effective flux errors in each band can vary substantially with position, depending on the local density of prior sources over areas comparable to the PSF. A detailed description of the flux measurements and Monte Carlo (MC) derivations of the uncertainties will be presented elsewhere (E. Daddi et al., in preparation). We correct the PACS photometry for a small flux bias introduced (due to source filtering for background subtraction) during data reduction (see H-GOODS public data release; Popesso et al. in preparation).

### 2.2. A Sample of Main Sequence Galaxies at z∼1.5 and z∼0.5

Daddi et al. (2010) presented PdBI CO[2-1] emission line detections of six star-forming galaxies at , originally selected by using the “star-forming BzK” color criterion (Daddi et al. 2004b). All six sources (BzK-21000, BzK-17999, BzK-4171, BzK-16000, BzK-12591 and BzK-22536) have spectroscopic redshifts (now confirmed by multiple CO detections; for optical redshifts see Stern et al. in preparation, Cowie et al. 2004) and robust PACS and/or SPIRE detections. Four of these sources also benefit from 1.3 mm continuum data derived as a by-product from the CO[5-4] emission line observations (Dannerbauer et al. 2012 in prep). For BzK-21000, continuum detections and uper limits are also obtained at 1.1, 2.2, and 3.3 mm (Daddi et al. 2010a; Dannerbauer et al. 2009). In addition to IRAC and MIPS 24m data, the sources are seen in the 16m InfraRed Spectrograph peak-up image (Teplitz et al. 2011). Although we will revisit (and confirm) the star formation rates of these sources based on the Herschel data, the existing UV, mid-IR and radio SFR estimates, along with stellar mass measurements derived by fitting the Bruzual & Charlot (2003) model SEDs to their rest-frame UV to near-IR photometry (Daddi et 2010a), consistently place them in the SFR main sequence at (Daddi et al. 2010). Finally, the UV rest-frame morphologies, the double-peaked CO profiles, the large spatial extent of their CO reservoirs, and the low gas excitation of the sources, all provide strong evidence that they are large, clumpy, rotating disks (Daddi et al. 2010a).

We also consider three normal disks with Sérsic index and spectroscopic redshifts, for which we have CO[2-1] emission line detections (Daddi et al. 2010b). Similar to the sample, the sources are detected in the PACS and/or SPIRE bands and are part of the SFR main sequence based on their 24m–derived IR luminosities and SFRs, which are known to be robust in this redshift range (Elbaz et al. 2010, 2011). The Herschel and millimetre photometry of the main sequence galaxies considered here are summarised in Table 1.

### 2.3. A Sample of High−z SMGs

As a comparison sample to our MS galaxies, we also include in our analysis a small sample of SMGs: GN20, SMMJ2135-0102 and HERMES J105751.1+573027. The selection of the targets was driven by the need for available multi-wavelength photometry, including Herschel and mm continuum observations, as well as CO[1-0] emission line detections.

GN20 was already studied in Magdis et al. (2011). It is one of the best-studied SMGs to date, the most luminous and also one of the most distant (, Pope et al. 2006, Daddi et al. 2009) in the GOODS-N field. Herschel photometry and the far-IR/mm properties of the source have already been presented in detail in Magdis et al. (2011b). In brief, the source is detected in all Herschel bands (apart from 100m) and in the AzTEC 1.1 mm map (Perera et al. 2008) while continuum emission is also measured at 2.2, 3.3, and 6.6 mm (Carilli et al. 2011) as well as at 1.4 GHz with the VLA (Morrison et al. 2010). Furthermore, Carilli et al. (2010) reported the detection of the CO[1-0] and CO[2-1] lines with the VLA, and CO[6-5] and CO[5-4] lines with the Plateau de Bure Interferometer (PdBI) and the Combined Array for Research in Millimeter Astronomy (CARMA), respectively. A compilation of the photometric data is given in Table 1 of Magdis et al. (2011b).

SMMJ2135-0102 (SMM-J2135 hereafter) is a highly magnified SMG at , with an amplification factor of = 32.4 4.5, serendipitously discovered by Swinbank et al. (2010) behind the cluster MACS J2135-01. The source has exquisite photometric coverage from rest-frame UV to radio including SPIRE broadband data (Swinbank et al. 2010 Table 1 and Ivison et al. 2010 Table 1). Swinbank et al. (2010) also report the detection of CO[1-0] and CO[3-2] emission lines using the Zpectrometer on the Green Bank Telescope and PdBI observations, while Ivison et al. (2010), using the SPIRE Fourier Transform Spectrometer (FTS), presented the detection of the [CII]158m cooling line.

Finally, HERMES J105751.1+573027 (HSLW-01 hereafter), is a Herschel/SPIRE-selected galaxy at z=2.957, multiply-lensed (magnification factor ) by a foreground group of galaxies. The source was discovered in Science Demonstration Phase Herschel/SPIRE observations of the Lockman-SWIRE field as part of the Herschel Multi-tiered Extragalactic Survey (HerMES; S. Oliver et al. 2012). The optical to mm photometry of the source is presented in Table 1 of Conley at al. (2011), while Riechers et al. (2011) reports the detection of CO[5-4], CO[3-2], and CO[1-0] emission using the PdBI and CARMA and the Green Bank Telescope.

We note that due to the high magnification factor for SMM-J2135 and HSLW-01, these sources might not be representative of the bulk population of SMGs, traditionally selected as mJy. Also, a large number of non-lensed SMGs with CO[3-2] observations is available in the literature. However, we choose not to consider them in this study, as the uncertain gas excitation results in dubious CO[1-0] estimates (e.g., Ivison et al. 2010, Riechers et al. 2011), which are essential for investigating the gas properties of the sources.

### 2.4. A Statistical Sample of z∼1 and z∼2 MS Galaxies

In addition to our samples of individually-detected MS galaxies at and , we also perform a stacking analysis to derive the average SED of MS galaxies at and . Specifically, we use the GOODS samples of Salmi et al. (2012; see also Daddi et al. 2007 and Pannella et al. in prep). In order to remove bulge-dominated galaxies with low sSFR, we discard objects with a Sérsic index (Salmi et al. 2012), as measured from the GOODS ACS images (Giavalisco et al. 2004), and consider only 24m detected galaxies from the GOODS ultra-deep Spitzer imaging. To remove starbursts we proceed as follows: i) from and redshift, we compute sSFR, i.e., the fiducial mean sSFR expected for MS galaxies with a given mass and redshift, derived as described at the start of §2 ii) from Herschel data we derive  and subsequently SFR estimates, and exclude galaxies with measured sSFR/sSFR 3. Since not all starburst systems are expected to be detected by Herschel we also omit sources with sSFRs/sSFR 3 using the UV-based, corrected for extinction, SFR estimates by Daddi et al. (2007) and the Spitzer estimates of Salmi et al. (2012). The sources are drawn from the BzK–selected sample of Daddi et al. (2007; now extended to deeper K-band depths by Pannella et al. in prep.). Since this color-selection technique naturally excludes quiescent galaxies, we only exclude starbursts identified using the same method as for the sample. The total number of and galaxies considered in the stacking analysis is 1569 and 3618, respectively. The average SEDs of these two samples are then measured by stacking, using various standard techniques depending on the wavelength:

• 16m: We use the Teplitz et al. (2011) maps. The stacking is performed using the IAS stacking library (Béthermin et al. 2010a) and PSF-fitting photometry.

• 24 and 70m: We use the 24m and 70m (Frayer et al. 2006b) maps of GOODS, the IAS stacking library and aperture photometry with parameters similar to that in Béthermin et al. (2010a).

• 100 and 160m: We use the GOODS-Herschel PACS data (Elbaz et al. 2011), the IAS stacking library and PSF-fitting. We apply the appropriate flux correction for faint, non masked, sources to the PACS stacks (Popesso et al. in preparation).

• 250, 350 and 500 m: We use the GOODS-Herschel SPIRE (Elbaz et al. 2011) data and the mean of pixels centred on sources as in Béthermin et al. (2012a). The bias due to clustering is estimated to be about 20% at 500 m and thus negligible compared with the statistical uncertainties. This bias is smaller at shorter wavelengths where the beam is narrower (Béthermin et al. 2012a).

• 870 m and 1100 m: We used the Weiß et al. (2009) LABOCA map of eCDFS and the Perera et al. (2008) AzTEC map of GOODS-N. Contrary to SPIRE, (sub-)mm data are noise-limited, it is thus optimal to beam-smooth the map before stacking (Vieira et al. in prep.). We thus apply the Marsden et al. (2009) method to perform our stacking, which takes the mean of pixels centered on sources in the beam-smoothed map.

At all wavelengths, we used a bootstrap technique to estimate the uncertainties (Jauzac et al. 2011), and the mean fluxes measured in the two fields are combined quadratically to produce a mean SED at and . For the normal galaxies we also construct average SEDs in three stellar mass bins in the range of with a bin size of , as well as in four bins of sSFR/sSFR over the range , with a bin size of .

## 3. Derivation of Far-IR Properties

Several key physical properties of distant galaxies, such as infrared luminosities (), dust temperatures () and dust masses (), can be estimated by fitting their mid-to-far-IR SEDs with various models and templates calibrated in the local Universe. However, the lack of sufficient data for a proper characterization of the SED of distant galaxies has often limited this kind of analysis to models suffering from (necessarily) over-simplified assumptions and large generalizations. The Spitzer, Herschel, and millimeter data available for the galaxies in our high sample provide thorough photometric sampling of their SEDs, allowing the use of more realistic models that previously have been applied mainly in the analysis of nearby galaxies. Here we consider both the physically-motivated Draine & Li (2007) (DL07 hereafter) models, as well as the more simplistic, but widely used, modified black body model (MBB).

### 3.1. The Draine & Li 2007 Model

We employ the dust models of DL07, which constitute an update of those developed by Weingartner & Draine (2001) and Li & Draine (2001), and which were successfully applied to the integrated photometry of the Spitzer Nearby Galaxy Survey (SINGS) galaxies (Draine et al. 2007). These models describe the interstellar dust as a mixture of carbonaceous and amorphous silicate grains, whose size distributions are chosen to mimic the observed extinction law in the Milky Way (MW), the Large Magellanic Cloud (LMC), and the Small Magellanic Cloud (SMC) bar region. The properties of these grains are parametrized by the PAH index, , defined as the fraction of the dust mass in the form of PAH grains. The majority of the dust is supposed to be located in the diffuse ISM, heated by a radiation field with a constant intensity . A smaller fraction of the dust is exposed to starlight with intensities ranging from to , representing the dust enclosed in photo-dissociation regions (PDRs). Although this PDR component contains only a small fraction of the total dust mass, in some galaxies it contributes a substantial fraction of the total power radiated by the dust. Then, according to DL07, the amount of dust  exposed to radiation intensities between and can be expressed as a combination of a -function and a power law:

with (),  the total dust mass, the fraction of the dust mass that is associated with the power-law part of the starlight intensity distribution, and characterizing the distribution of starlight intensities in the high-intensity regions.

Following DL07, the spectrum of a galaxy can be described by a linear combination of one stellar component approximated by a blackbody with color temperature = 5000K, and two dust components, one arising from dust in the diffuse ISM, heated by a minimum radiation field (“diffuse ISM” component), and one from dust heated by a power-law distribution of starlight, associated with the intense photodissociation regions (“PDR” component). Then, the model emission spectrum of a galaxy at distance is:

 fmodelν=Ω∗Bν(T∗)+Mdust4πD2[(1−γ)p(0)ν+γpν], (2)

where is the solid angle subtended by stellar photospheres, and are the emitted power per unit frequency per unit dust mass for dust heated by a single starlight intensity , and dust heated by a power-law distribution of starlight intensities extending from to .

In principle, the dust models in their most general form are dictated by seven free parameters, ( and ). However, Draine et al. (2007) showed that the overall fit is insensitive to the adopted dust model (MW, LMC and SMC) and the precise values of and . In fact they showed that fixed values of and successfully described the SEDs of galaxies with a wide range of properties. They also favor the choice of MW dust models for which a set of models with ranging from 0.4% to 4.6% is available. Furthermore, since small values correspond to dust temperatures below K that cannot be constrained by far-IR photometry alone, in the absence of rest-frame submm data, they suggest using 0.7 25. While this lower cutoff for prevents the fit from converging to erroneously large amounts of cold dust heated by weak starlight ( 0.7), the price to pay is a possible underestimate of the total dust mass if large amounts of cold dust are indeed present. However, Draine et al. (2007) concluded that omitting submm data from the fit increases the scatter of the derived masses up to 50% but does not introduce a systematic bias in the derived total dust masses.

Under these assumptions, we fit the mid-IR to mm data points of each galaxy in our sample, searching for the best fit model by minimization and parametrizing the goodness of fit by the value of the reduced , (where is the number of degrees of freedom). The best fit model yields a total dust mass (), , and while to derive 111 quoted below is similar to , but integrated from 0 to  estimates we integrate the emerging SEDs from 8- to 1000m:

 LIR=∫1000μm8μmLν(λ)×cλ2 dλ. (3)

A by-product of the best fit model is also the dust weighted mean starlight intensity scale factor, , defined as:

 ⟨U⟩=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩LdustP0Mdust      or[(1−γ)Umin+γln(Umax/Umin)U−1min−U−1max],   for α=2 (4)

where is the power absorbed per unit dust mass in a radiation field with . Note that essentially is proportional to /, and as we will discuss later for the definition of  adopted here, i.e. , our data suggest that 125. Uncertainties in  and  are quantified using Monte Carlo simulations. To summarise, for each galaxy a Gaussian random number generator was used to create 1000 artificial flux sets from the original fluxes and measurement errors. These new data sets were then fitted in the same way, and the standard deviation in the new parameters was taken to represent the uncertainty in the parameters found from the real data set. Best fit values along with their corresponding uncertainties are listed in Table 2 for all sources in our sample. To check for possible contamination of the submm continuum broad band photometry from C+ (158m) emission (e.g., Smail et al. 2011), we repeated the fit excluding the affected bands (i.e., 350m and 500m for 0.5 and respectively without noticing any effect in the derived parameters.The best fit models along with the observed SEDs of the , galaxies and SMGs are shown in Figures 1, 2 and 3 respectively.

### 3.2. The Importance of Millimeter Data

While Herschel data accurately probe the peak of the SED in the far-IR emission of distant galaxies, rest-frame submm observations (m) are necessary to sample the Rayleigh-Jeans tail. Since the available far-IR photometry for five of our sources is restricted to Herschel observations, it is important to explore possible biases or systematics introduced by the lack of mm continuum data. Given the ever increasing number of distant galaxies with Herschel photometry, this investigation will also provide guidance for similar studies in the future. To assess the significance of adding mm photometry in the derivation of the far-IR properties of the galaxies, and particularly their dust mass, we repeat the fitting procedure, this time excluding any data at wavelengths longer than rest-frame 200m, for the four BzK and the three SMGs with available mm continuum observations.

A comparison between the derived dust masses with and without the use of mm data is shown in Figure 4 (left). We find no evidence for a strong systematic bias, with an average ratio between the two estimates . Interestingly, the sources with the largest discrepancies are the SMGs, although the sample is too small to clearly demonstrate the existence of a systematic effect. However, the addition of mm data has a noticeable impact on the uncertainties of the derived  estimates, which are reduced, on average, by a factor of . Studies in the local universe reach similar conclusions, with the addition that in the absence of rest-frame sub-mm data, dust masses tend to be underestimated for metal-poor galaxies. This could serve as an indirect indication that our sample mainly consists of metal-rich sources, something that we will also argue in Section 4.

The derived and are also in broad agreement between the two cases. However, we notice a weak systematic bias towards higher values, i.e., stronger mean radiation fields, when mm data are considered in the fit, reflecting the fact that /. We conclude that such trends in and  suggest that mm data can place better constraints on the diffuse ISM emission, as well as on the relative contribution of a PDR component to the total radiation field, and consequently on the  of individual high galaxies. Finally, we note that the detailed sampling of the peak of the far-IR emission provided by Herschel data can result in robust  estimates without the need of mm data.

### 3.3. Comparison With Modified Black Body Fits

Another method for deriving estimates for the dust masses and other far-IR properties such as dust temperatures and dust emissivity indices (), is to fit the far-IR to submm SED of the galaxies with a single-temperature modified blackbody, expressed as:

 fν∝ν3+βehνkTd−1 (5)

where  is the effective dust temperature and is the effective dust emissivity index. Then, from the best fit model, one can estimate  from the relation:

 Md=SνD2L(1+z)κrestBν(λrest,Td),  with  κrest=κ0(λoλrest)β (6)

where is the observed flux density, is the luminosity distance, and is the rest-frame dust mass absorption coefficient at the observed wavelength. While this is a rather simplistic approach, mainly adopted due to the lack of sufficient sampling of the SED of distant galaxies, it has been one of the most widely-used methods in the literature. Therefore, an analysis based on MBB-models not only provides estimates of the effective dust temperature and dust emissivity of the galaxies in our sample, but also a valuable comparison between dust masses inferred with the MBB and DL07 methods.

We fit the standard form of a modified black body, considering observed data points with , to avoid emission from very small grains that dominate at shorter wavelengths. For the cases where mm data are available we let vary as a free parameter, while for the rest we have assumed a fixed value of , typical of disk-like, main sequence galaxies (Magdis et al. 2011b, Elbaz et al. 2011). From the best fit model, we then estimate the total  through equation 6. For consistency with the DL07 models we adopt a value of = 5.1 cm g (Li & Draine 2001). To obtain the best fit models and the corresponding uncertainties of the parameters, we followed the same procedure as for the DL07 models. The derived parameters are summarized in Table 2 and the best fit models are shown in Figures 1, 2 and 3.

A comparison between dust masses derived by DL07 and MBB models in shown in Figure 4 (right). We see that a modified black body appears to infer dust masses that are lower than those derived based on DL07 models on average by a factor of (). A similar trend was recently reported by Dale et al. (2012), who found that the discrepancy between the two dust mass estimates is smaller for sources with warmer colors. Interestingly, when we convolve the best fit DL07 rest-frame SEDs of our galaxies with the 70- and 160m PACS filters, we find that the sources with the warmest are indeed those with the best agreement in the dust masses derived from the two methods. The reason behind this discrepancy has been addressed by Dunne et al. (2000). The grains of a given size and material are exposed to different intensities of the interstellar radiation field and thus attain different equilibrium temperatures which will contribute differently to the SED. The single-temperature models cannot account for this range of  in the ISM and attempt to simultaneously fit both the Wien side of the grey body, which is dominated by warm dust, as well as the Rayleigh-Jean tail, sensitive to colder dust emission. The net effect is that the temperatures are driven towards higher values, consequently resulting in lower dust masses. While restricting the fit to m does not provide an adequate solution to the problem (Dale et al. 2012), a two-temperature black body fit is more realistic and returns dust masses that are larger by a factor of compared to those derived based on a single  MBB (Dunne & Eales 2001), in line with those inferred by the DL07 technique.

Despite the large uncertainties associated with the MBB analysis, it is worth noting that for all SMGs in our sample we measure an effective dust emissivity index , in agreement with the findings of Magnelli et al. (2012). On the other hand, the far-IR to mm SEDs of all BzK galaxies are best described by , similar to that found by Elbaz et al. (2011) for MS galaxies. However, we stress that the effective does not necessarily reflect the intrinsic of the dust grains due to the degeneracy between and the temperature distribution of dust grains in the ISM. The - degeneracy also prevents a meaningful comparison between the derived  for the galaxies in our sample.

## 4. The Gas Content of Main Sequence and Star-bursting Galaxies

Several studies in the local Universe have revealed that the total gas-to-dust mass ratio () is correlated with the gas-phase oxygen abundance, in the sense that more metal-rich galaxies tend to exhibit lower  values (e.g., Muñoz-Mateos et al. 2009, Leroy et al. 2011 and references within). In addition to crucial information regarding the amount of metals trapped in the dust, this correlation can serve as a valuable tool for deriving indirect estimates of the CO to molecular gas mass ()  conversion factor. In particular, if we know the metallicity, the dust mass and the mass of the atomic gas mass () of a galaxy, then we can estimate the  through the relation, and  through the following relation:

 δGDRMdust≡Mgas=MH2+MHI (7)

Subsequently, if  is measured then we can estimate , since =   . This method has successfully been applied in the local Universe (e.g., Leroy et al. 2011) and has recently been extended to high galaxies by the pilot study of Magdis et al. (2011b). Even though restricted to only two galaxies, which are also part of this study (GN20 and BzK21000), Magdis et al. (2011b) showed that this approach provides  estimates that are consistent with other independent measurements and dynamical arguments. Although it has been a common practice to adopt a value of  = 0.8, typical of local ULIRGs, for high ULIRGs, our earlier study suggested that such low values apply only to the subset of genuinely star-bursting systems at high redshift, while high main sequence galaxies, even those with (i.e., those classified as ULIRGs) have larger  values, similar to those of local spirals. Here, we wish to extend this experiment, this time using a substantially larger sample, and to investigate possible correlations between  and starburst versus main sequence indicators.

### 4.1. Estimating Metallicities

A key ingredient of this method is the metallicity of the galaxies, for which we have to rely on indirect measurements. We first derive stellar mass estimates by fitting the Bruzual & Charlot (2003) model SEDs to their rest-frame UV to near-IR spectrum. Then for the BzK galaxies and SMMJ2135-0102 we use the relation at from Erb et al. (2006), while for the sources we adopt the fundamental metallicity relation (FMR) of Mannucci et al. (2010) that relates the SFR and the stellar mass to metallicity. To check for possible systematics, we also derive metallicity estimates for the sources with the FMR relation and find that the two methods provide very similar results. For GN20 and HLSW-1 the situation is more complicated, as the FMR formula is only applicable up to , and the Erb et al. (2006) relation saturates above . For these two sources we adopt the line of reasoning of Magdis et al. (2011b). Namely, in addition to the metallicity estimates based on the FMR relation, we also consider the extreme case where the huge SFR of the two sources (  yr) is due to a final burst of star formation triggered by a major merger that will eventually transform the galaxy into a massive elliptical. Once star formation ceases, the mass and metallicity of the resulting galaxy will not change further, and one might therefore apply the mass–metallicity relation of present-day elliptical galaxies (e.g., Calura et al. 2009). Combining the metallicity estimates based on this scenario with those derived based on the FMR relation, we estimate that the metallicities could fall in the ranges 8.8 to 9.2 for GN20, and 8.6 to 9.0 for HLSW-1. For the analysis we will adopt for GN20 and for HLSW-1. We also adopt a typical uncertainty of 0.2 for the rest of the sources. The assumed metallicities, all calibrated at the Pettini & Pagel (2004) (PP04) scale, along with the stellar masses, are given in Table 3.

### 4.2. Derivation of the CO to MH2 Conversion Factor

To derive  estimates for our sample, we first derive a relation using the local sample presented by Leroy et al. (2011), after converting all metallicities to the PP04 scale (Figure 5 left). The data yield a tight correlation between the two quantities555Adopting the relation quoted by Leroy et al. (2011), i.e.,  , has virtually no impact on the results presented here, with a mean difference between the derived  estimates of a factor of described as:  , with a scatter of 0.15 dex. Having obtained the  and metallicity estimates, we then use the relation to derive  and subsequently estimate  from the equation  = . For the last step we have assumed that at high,   or equivalently that   , as supported by both observational evidence (e.g., Daddi et al. 2010; Tacconi et al. 2010, Geach et al. 2011) and theoretical arguments (Blitz & Rosolowsky 2006; Bigiel et al. 2008, Obreschkow et al. 2009). All values and their corresponding uncertainties, which take into account both the dispersion of the relation and the uncertainties in and , are summarized in Table 3. In Figure 5 (right), we plot the derived  values as a function of metallicity, along with the estimates of Leroy et al. (2011) for a sample of local normal galaxies as well as for a sample of local ULIRGs from Downes & Solomon (1998), for which we were able to compute their metallicities on the PP04 scale. Fitting high MS galaxies along with the local sample of Leroy et al. (2011), yields:

 log(αCO)=−(1.39±0.3)×(12+log[O/H])PP04+12.8±2.2 (8)

with  decreasing for higher metallicities, in agreement with previous studies (e.g., Wilson 1995, Israel et al. 1997, Schruba et al. 2012, and references within). It is evident that main sequence galaxies at any redshift have higher  values than those of local star-bursting ULIRGs and more similar to those of local normal galaxies. In particular, for high MS galaxies, we find a mean  of 5.50.4, very close to the MW value of 4.6 (e.g., Strong & Mattox 1996, Dame et al. 2001). Interestingly, SMM-J2135 falls close to the locus of main sequence galaxies with  , while for the remaining SMGs we find  , close to the average value of local ULIRGs ( , e.g., Solomon et al. 1997, Tacconi et al. 2008). Finally, simply as a consistency check, we derive and plot in Figure 5 (left) the / values of our targets, using  estimates as obtained from equation 8.

### 4.3. Limitations

Before trying to interpret our findings regarding the  values, it is important to discuss possible caveats and limitations of the method. A key ingredient for the derivation of  is the  estimate. These estimates rely heavily on the adopted model of the grain-size distribution. Dust masses based on different grain size compositions and opacities in the literature can vary even by a factor of , indicating that the absolute values should be treated with caution (e.g., Galliano et al. 2011). However, the relative values of dust masses derived based on the same assumed dust model should be correct and provide a meaningful comparison, as long as the dust has similar properties in all galaxies. Therefore, any trends arising from the derived  estimates are essentially insensitive to the assumed dust model.

One important assumption of our technique is that the relation defined by local galaxies does not evolve substantially with cosmic time. These quantities are very intimately related, as dust is ultimately made of metals. For example, solar metallicity of = 0.02 means that 2% of the gas is in metals. The local relation suggests an average trend where 1% of the gas into dust. This means that about 50% of the metals are locked into dust. This is already quite an efficient process of metal condensations into dust, and it is hard to think how this could be even more efficient at high- (where, e.g., less time is available for the evolution of low-mass stars into the AGB phase), with a much higher proportion of metals locked into the dust phase, a situation that would lead to an overestimate of with our technique. Similarly, chemical evolution studies suggest a small evolution of dust-to-metal ratio, i.e., by a maximum factor of 2 from 2 Gyr after the Big Bang up to the present time (e.g., Edmunds 2001, Inoue 2003, Calura et al. 2008). Furthermore, if we plot the ratio of the direct observables, /, as a function of metallicity, both local and high galaxies follow the same trend, suggesting that the local relation is valid at high (Figure 6). Finally, while the Leroy et al. (2011) sample yields a slope of in the relation, Muñoz-Mateos et al (2009), find a rather steeper slope of -2.45. However, adopting the Muñoz-Mateos relation (after rescaling it to PP04 metallicity), returns very similar  values, as the two relations intercept at metallicities very close to the average metallicity of our sample (). We note though that for GN20 and HLSW (the sources with the highest Z estimates), the Muñoz-Mateos relation would reduce  values by a factor of .

### 4.4. αCO as a Function of Specific Star Formation Rate

With robust  measurements in hand, we are in a position to derive accurate estimates of the SFR and sSFR for galaxies in our sample. We first convert the derived to star formation rates by using the Kennicutt (1998) relation, scaled for a Chabrier IMF for consistency with our stellar mass estimates. We then infer the specific star formation rate of each source and compare it to the characteristic sSFR of the main sequence galaxies at the corresponding redshift of the source, using the methodology described in §2 up to . For the two SMGs at and , we use the SFR- relations of Magdis et al. (2010a) and Daddi et al. (2009) respectively. In order to conservatively identify MS galaxies, we classify starbursts as galaxies with an excess sSFR relative to that of the MS galaxies by at least a factor of 3, i.e., with sSFR/sSFR 3. We note however that the sSFR/sSFR indicator is only a statistical measure of the star formation mode of the galaxies, and there is no rigid limit that separates main sequence from starburst galaxies.

Our analysis suggests that high MS galaxies, associated with a secular star formation mode, exhibit higher  values at any redshift compared to those of merger-driven local ULIRGs. Since the latter are known to be strong outliers from the local SFR-M relation, we expect a dependence of the  value with sSFR. However, the functional dependence between the two quantities is not straightforward. In the local universe, there is evidence for a bimodality in the  value, linked with the two known star formation modes: normal disks (for ) are associated with   4.4 and merger driven ULIRGs with   0.8. As we will discuss later, if the  of normal galaxies depends only on metallicity, then we expect only a weak increase of  as a function of sSFR/sSFR within the MS. Consequently, in order to incorporate the observed steep decrease of  in starburst systems,  and the relative offset from the main sequence (i.e., sSFR/sSFR) will be related with a step-function. On the other hand, if  also depends on other parameters than metallicity, such as the compactness and/or the clumpiness of the ISM of normal galaxies, and if the ISM conditions of MS galaxies do vary significantly, then  could smoothly decrease as we depart from the MS, and move towards the starburst regime (e.g., Narayanan et al. 2012).

The scenario of a continuous variation of  with sSFR/sSFR is depicted in Figure 7a. In addition to our galaxies we also include a sample of local normal galaxies for which we have robust and SFR estimates from Leroy et al. (2008) and  measurements from Schruba et al. (2012). A linear regression fit yields the following relation:

 αCO=(5.8±2.0)×[sSFR/sSFRMS](−0.67±0.25) (9)

with a scatter of 0.3 dex. We stress that, given the lack of a sufficient number of starbursts in our sample, this relation is poorly constrained in the starburst regime. Nevertheless, the locus of local ULIRGs, seems to be consistent with the emerging trend. Note that since we lack a sample of local ULIRGs for which both  and sSFR are accurately determined, the position of local ULIRGs in Figure 7 indicates the average  and sSFR of the population, as derived by Downes & Solomon (1998) and Da Cunha et al. (2010b), respectively. To further check the robustness of the fit we also perturbed the original the data within the errors and repeated the fit for 1000 realisations. The mean slope derived with this method is very close to the one describing the original data. A Spearman’s rank correlation test yields a correlation coefficient of , with a value of 0.03, suggesting a moderately significant correlation between sSFR/sSFR and . However, repeating the Spearman’s test, this time excluding star-bursting systems, does not yield a statistically significant correlation, with and a value of 0.3, indicating a very small dependence, if any, between  and sSFR within the MS.

It is therefore unclear whether  varies inside the MS sequence, or if instead there is a bifurcation between the mean values appropriate for MS versus SB galaxies. If the  conversion factor primarily depends on metallicity, one would expect little or no variations with sSFR at constant mass within the MS. Detailed computations for this scenario are presented in Sargent et al. (2012b, in prep.), in which burst-like activity is assumed to be characterised by a constant, low conversion factor, chosen here to be = 0.8. To predict the variations of as a function of sSFR/sSFR according to this scenario, Sargent et al. (2012) compute the relative importance of main-sequence and starburst activity at a given position within the stellar mass vs. SFR plane using the decomposition of the sSFR-distribution at fixed stellar mass derived in Sargent et al. (2012a; based on the data of Rodighiero et al. 2011). Metallicities, which form the basis for computing for an ISM experiencing “normal” star-formation activity, vary smoothly as a function of stellar mass and star formation rate according to the calibration of the fundamental metallicity plane given in Mannucci et al. (2010; see also Lara-Lopez et al. 2010). To compute the corresponding value of the conversion factor, Sargent et al. (2012b in prep.) assume a variation of as they find that this relation between metallicity and reproduces the faint-end slope of the CO-LF of Keres et al. (2003) best, under the set of assumptions just described.

The colored tracks in Figure 7b delineate the variations of with sSFR/ expected for galaxies in three bins of total stellar mass. Three regions can be clearly distinguished: (i) the main-sequence locus, characterised by a gradual increase of with sSFR that is caused by the slight rise in metallicity predicted by the FMR; (ii) the starburst region at high sSFR/, where the conversion factor, by construction, assumes a mass- and redshift-independent, constant value; and (iii) a narrow transition region (spanning roughly sSFR/) where drops from an approximately Milky Way-like conversion factor to the starburst value. Note that, in contrast to the previous case,  varies only slightly among MS galaxies. This weak dependence of  on sSFR/sSFR introduces a step as we move from MS to starburst systems. In the emerging picture, the majority of the star-forming population is dominated by either the main-sequence or starburst mode, and only a small fraction consists of “composite” star-forming galaxies hosting both normal and starburst activity. For composite sources, the should be interpreted as a mass-weighted, average conversion factor that reflects the relative amount of the molecular gas reservoir that fuels star-forming sites experiencing burst-like and secular star-formation events, respectively. Note also that variations with redshift (indicated by different colored lines in Figure 7b) are small for the limited range of stellar mass covered by our sample, and would likely be indistinguishable within the natural scatter about the median trends plotted in the Figure.

Both scenarios appear to be consistent with the data, leaving open the question of whether the transition from normal to starburst galaxies is followed by a smooth or a step-like variation of . However, both scenarios agree in that  should not vary much within the main sequence. Indeed, even for the case of continuous variation, the decrease of  becomes statistically significant only for strongly star-bursting systems, as equation 9 indicates that  would vary at most by a factor of within the full range of the MS, well within the observed scatter of the relation. Therefore, the average value  , as derived in this study, can be regarded as representative for the whole population of MS galaxies at any redshift, with stellar masses . We remind the reader that because sSFR increases with redshift as (1+z), at least out to , galaxies with (U)LIRG-like luminosities can enter the MS at higher redshifts or at large stellar masses (e.g., Sargent et al. 2012). We stress that our analysis suggests that the  of these systems, i.e., high main sequence ULIRGs, is on average a factor of 5 larger than that of local ULIRGs.

### 4.5. Comparison with Theoretical Predictions

The determination of the  value has also been the focus of several theoretical studies. In particular Narayanan et al. (2011), investigated the dependence of  on the galactic environment in numerical simulations of disk galaxies and galaxy mergers, and reported a relationship between  and the CO surface brightness of a galaxy. Here, we compare our observationally constrained  values with those predicted by their theoretical approach. According to Narayanan et al. (2011) :

 αCO=6.75×1020⟨WCO⟩−0.326.3×1019×(Z/Z⊙)0.65 (10)

where , is the CO surface brightness of the galaxy in K Km s. According to Solomon & Vanden Bout (2005)

 L′CO=23.5 × ΩB∗S D2L I′CO (1+z)−3 (11)

where  is measured in K km s pc, is the solid angle of the source convolved with the telescope beam measured in arcsec, and I is the observed integrated line intensity in K km s. In the case of an infinitely good resolution, as assumed in Narayanan’s simulations, this becomes the solid angle subtended by the source. The CO line luminosity can also be expressed for a source of any size in terms of the total line flux:

 L′CO=3.25×107 SCO Δv ν−2obs D2L (1+z)−3 (12)

where v is the velocity integrated flux in Jy km s. From equations 11 and 12, and for the case of an infinitely good resolution, we derive :

 WCO=1.38×106 4× ν−2obsπ(1+z) θ−2 ×SCOΔv (13)

where is the observed frequency and is the size of the source in arcsec666We introduce a term here, not present in the original Narayanan et al. formula, which is required in order to make independent on redshift for fixed physical properties of the galaxies. Without that term the systematic difference between predicted and measured values would grow to a factor of 2. For our sources we use, where possible, the CO sizes, and UV sizes when CO observations are too noisy to reliably measure the source extent. Using equations 10 and 13, we then get an estimate of  for each source in our sample according to the prescription of Narayanan et al. (2011), and in Figure 8 we compare them to the values derived based on our method. Although there seems to be a small systematic offset (on average by a factor of 1.4) towards lower values for the theoretical approach, the overall agreement of the  values is very good when one considers the large uncertainties and assumptions of the two independent methods. We stress that the derived and therefore the estimated  values are very sensitive to the choice of the size indicator.

### 4.6. Implications for the Star Formation Activity of High−z MS Galaxies

While the sSFR probes the star formation mode of a galaxy only indirectly and from a statistical point of view, accurate measurements of its star formation efficiency, SFE = SFR/, can provide more direct and reliable answers. With the robust  estimates derived in this study, we can convert the CO measurements into  and subsequently, infer the star formation efficiency of galaxies in our sample. In what follows we use SFE = SFR/  and SFE = / interchangeably, since SFR is linearly related to  through SFR   (i.e., through the Kennicutt 1998 relation, calibrated for a Chabrier IMF).

We have so far established that even though there appears to be only small variation of  among MS galaxies, the  of high MS galaxies is similar to that of local disks with lower , and is approximately 5 times larger than the average  value observed for local ULIRGs. The direct implication of this finding is that for a given , one could expect a large variation in the amount of molecular gas for galaxies between MS and starburst galaxies. This also suggests that for a given  (or equally, SFR), galaxies in the main sequence have lower star formation efficiencies, as compared to star-bursting systems. Indeed, using the inferred  values we derive SFE estimates for each source in our sample and plot them against sSFR/sSFR in Figure 9a. We supplement our sample with a compilation of local disks (Leroy et al. 2008), as well as with a sample of local ULIRGs (Solomon et al. 1997, Rodríguez Zaurín et al., 2010).

Although the actual dependence of SFE on sSFR/sSFR within the MS and the transition from MS to the starburst regime will be discussed in detail in the next section, here we can derive some crucial results about the SFE of the two populations (MS and starbursts galaxies) by employing a simple, empirical relation between SFE and sSFR/sSFR. Indeed, a linear regression fit and a Spearman’s test to our high data (including starbursts), yields a strong and statistically significant correlation ( = 0.71, -value = 0.0012) between star formation efficiency and sSFR/sSFR, with:

 SFE=(8.1±1.2)×[sSFR/sSFRMS]1.34±0.13 (14)

suggesting substantially higher star formation efficiencies for galaxies with enhanced sSFR (Figure 9a). For MS galaxies we find an average  L/, corresponding to an average gas consumption timescale ( ) of 0.7 Gyrs, indicative of long-lasting star formation activity. This is in direct contrast to the short-lived, merger-driven starburst episodes observed in local ULIRGs, with an average L/, times higher than that of MS galaxies. Since the majority of galaxies at any redshift are MS galaxies (e.g., Elbaz et al. 2011) and starbursts galaxies seem to play a minor role in the star formation density throughout the cosmic time (e.g., Rodrighero et al. 2011, Sargent et al. 2012), our results come to support the recently emerging picture of a secular, long lasting star formation as the dominant mode of star formation in the history of the Universe.

To ensure that our result is not an artifact of the assumed  values, in Figure 10 we also plot the ratio of the direct observables, / versus the offset from the main sequence for the same set of objects, omitting any assumptions for . We find the two values to strongly correlate with and a value of 3.4 and a functional relationship of:

 LIR/L′CO=34.6(±5.0)×[sSFR/sSFRMS]0.80±0.1 (15)

with MS galaxies exhibiting lower / values by a factor of . We note, however, that the absence of a well defined sample of high starbursts is striking. SMGs, which are frequently regarded as prototypical high starbursts, have recently turned out to be more likely a mixed ensemble of objects, with a significant fraction of them exhibiting star formation rates and star formation efficiencies typical of MS galaxies (Ivison et al. 2011, Rodighiero et al. 2011). Indeed, while we have only three SMGs in this study, we seem to face the same situation. Of the three SMGs considered here, only HSLW-01 appears to be a strong starburst, with sSFR/sSFR 10 and SFE 300 L/. GN20 is only marginally outside the MS regime, with sSFR/sSFR, although with high SFE  L/, while SMM-J2135 behaves like a MS galaxy with sSFR/sSFR 2.5 and SFE 30 L/, much lower than the star formation efficiency observed for local ULIRGs ( 100 L/). Clearly a larger, objectively selected sample of high starbursts is essential to improve this investigation.

## 5. Variations of Sfe Within the Main Sequence

In the previous section, we demonstrated that the bulk of MS galaxies exhibit higher   values and have lower star formation efficiencies than starburst galaxies. Here we will attempt to investigate possible variations of SFE within the MS, taking into account that the thickness, i.e., the spread of the SFR- correlation as traced by normally star-forming galaxies at any redshift, is not just an artifact produced by random noise, but a manifestation of the variation of the physical properties of the main sequence galaxies, such as color and clumpiness (e.g., Salmi et al. 2012; see also Elbaz et al. 2011). Since the actual quantity that drives a galaxy above or below the main sequence is yet unknown, we will consider two limiting scenarios, where the relative position of a galaxy with respect to the MS is driven by i) variations in the gas fraction ( = , or equally in the gas to stellar mass ratio, /)  while the star formation efficiency remains roughly constant within the main sequence, or ii) variations in the star formation efficiency of MS galaxies, while  remains constant, in order to explain the observed dispersion in the SFR plane. Indeed, in practice there are two ways that a MS galaxy could have higher SFR: either it has more raw material () to produce stars, or for the same amount of  it is more efficient at converting that gas into stars. The two scenarios have direct implications on the star formation law (SF-law) in the galaxies. The first case implies the existence of a global  –  relation that would apply to all MS galaxies, irrespective of their sSFR, like the one presented by Daddi et al. (2010b) and Genzel et al. (2010). On the other hand, the second scenario would imply variations of the star formation law, with parallel  –  relations with constant slope, but with a normalisation factor that strongly increases with offset from the MS. While a combination of the two possibilities might also be plausible, we will consider the two limiting cases for simplicity.

### 5.1. Limiting Cases

#### 5.1.1 Scenario I: a Global SF-Law for MS Galaxies; fgas as a Key Parameter

In this scenario, the physical parameter that drives the sSFR of a main sequence galaxy is the gas fraction, while the star formation efficiency remains roughly constant. In this case we have an SF law in the form:

 logMgas=ξgas×logSFR+C1, (16)

where = 1 would imply /SFR = constant. However, there is observational evidence that 0.8, i.e., slightly lower than unity (e.g., Daddi et al. 2010, Genzel et al. 2010, Sargent et al. 2012 in prep.), suggesting a very mild dependence of SFE from SFR (or from sSFR/sSFR, for fixed stellar mass,) with a slope of . On the other hand, dividing equation 16 by stellar mass indicates that / would vary as a function of sSFR (or equally as a function of SFR at fixed stellar mass) with:

In Figure 9b we explore the variations of SFE with normalised sSFR for three representative bins of mass and redshift, as predicted by Sargent et al. (2012 in prep.), based on the framework described above. The two factors that contribute to the evolutionary tracks are (i) the observation that at fixed SFR (or ) starbursts display more than an order of magnitude higher SFE than that of “normal” galaxies (e.g., Daddi et al. 2010b, Genzel et al. 2010, this study), and (ii) the fact that the dependence of SFR on  is slightly supra-linear, such that SFE increases with  even when the main-sequence galaxies and starbursts are considered individually (i.e., eq. 16). Point (ii) is reflected in the weak increase of SFE throughout the main-sequence and starburst regime (sSFR/ and sSFR/, respectively), while the rapid rise in SFE in the transition region is the manifestation of point (i) above. Note that the “jump” in SFE, from the normal to the starburst galaxies is not forced by the assumption of a bimodal star formation activity but naturally results from the weak increase of SFE with sSFR/sSFR within the MS. Similar tracks are presented in Figure 10, considering / instead of /.

#### 5.1.2 Scenario II: a Varying SF-Law for MS Galaxies; SFE as a Key Parameter

This scenario assumes that  for MS galaxies remains constant, and that the physical parameter that dictates the position of the galaxy on the SFR plane is the star formation efficiency. Since,  = const, then expressing SFE as a function of sSFR/sSFR, i.e., at fixed stellar mass, we have:

 logSFE=log(SFR)+C2 (18)

suggesting a linear dependence of SFE on sSFR/sSFR, in reasonable agreement with the slope derived by a linear fit (, eq. 14). Although a similar slope is found when the fit is redone for MS galaxies alone, excluding the starbursts, a Spearman’s test suggests that the correlation is statistically insignificant ( = 0.58, value = 0.13), leaving open the question of whether the trend is valid within the MS, or if it is driven solely by the starbursts. A similar picture emerges when, instead of , we consider the directly observable quantity  (Figure 10). Again, in this case, excluding the starburst data results in a statistically insignificant correlation. If indeed the star formation efficiency of MS galaxies follows such a steep increase as a function of sSFR/sSFR, aside the existence of parallel  -  SF-laws, it would also imply a smooth, continuous transition from MS to starbursts galaxies.

Both scenarios considered here appear to be consistent with our data, and we cannot formally distinguish between them based on these individual objects. However, as we will see in the next section, the shape of the SED, both for individually elected sources as well as for populations averaged via stacking, can provide more definite answers.

## 6. Dissecting the “Thickness” of the Main Sequence

So far, we have seen that the high-quality SEDs offered by Herschel and mm continuum data can be used to reveal a strong dependence of the  and SFE of a galaxy on its deviation from the main sequence (eq.9 and eq.14), highlighting significant differences between high MS galaxies and local ULIRGs, even though the former may have (U)LIRG-like luminosities and SFRs. The well-sampled SEDs considered in this study can provide the tools to further explore the implications of the two scenarios presented in the previous section.

We have argued that the relation / , observed in the local universe, holds at high. A direct consequence of this assumption is that the ratio between the infrared luminosity and the dust mass of a galaxy at any redshift would be:

 LIRMdust∝LIRMgas×Z=SFEZ (19)

with   SFR for dusty galaxies where the extinction is large.777In detail,  , with SFR becoming significant in the case of low extinction, hence typically at very low galaxy masses and metallicities or at high redshifts. Based on this relation, we can make some predictions for the different trends between / and and sSFR/sSFR, for the two scenarios described above. Afterwards, we will attempt to constrain the observational trend by stacking large samples of mass-selected high-z galaxies.