THE ALMA SPECTROSCOPIC SURVEY IN THE HUBBLE ULTRA DEEP FIELD: MOLECULAR GAS RESERVOIRS IN HIGH-REDSHIFT GALAXIES

The details of the ALMA data set (observations and data reduction) are presented in Paper I of this series. Here we briefly summarize the observational details that are relevant for the present study. The data set consists of two frequency scans in ALMA band 3 (3 mm, 84–115 GHz) and in band 6 (1 mm, 212–272 GHz). In the case of the 3 mm observations, we obtained a single pointing centered at R.A. = 03:32:37.900, decl. = –27:46:25.00 (J2000.0), close to the northern corner of the Hubble eXtremely Deep Field (XDF, Illingworth et al. 2013). The primary beam has a diameter of $\approx 65^{\prime\prime}$ at the central frequency of the band (99.5 GHz). The typical noise rms is ∼0.18 mJy beam⁻¹ per 50 km s⁻¹ channel. The 1 mm observations consist of a seven-pointing mosaic covering approximately the same area as the 3 mm observations. The typical noise rms is 0.44 mJy beam⁻¹ per 50 km s⁻¹ channel. The resulting 1 mm continuum image reaches a noise rms of 12.7 μJy beam⁻¹ at the center of the mosaic (see Paper II).

We complement the ALMA data with X-ray-to-far-IR photometry from public catalogs of this field, as well as optical/near-IR spectroscopic information where available. The main sources for the photometry are the compilations by Coe et al. (2006) and Skelton et al. (2014). The former includes optical photometry based on the original Hubble Space Telescope (HST) Advanced Camera for Survey (ACS) images of the Hubble UDF (Beckwith et al. 2006) and near-IR images obtained with HST NICMOS. The latter also compiles optical/near-IR observations with HST ACS and the Wide Field Camera 3 (WFC3) from the Hubble XDF (Illingworth et al. 2013), Spitzer IRAC, as well as a wealth of ground-based optical/near-IR observations. Spitzer MIPS data at 24 and 70 μm, as well as Herschel PACS and SPIRE data come from the work by Elbaz et al. (2011). X-ray data are taken from the Extended Chandra Deep Field South Survey (Lehmer et al. 2005) and from the Chandra Source Catalog (Evans et al. 2010).

Photometric redshifts (${z}_{\mathrm{photo}}$) are available for all of the optically selected sources in the field. At a limiting magnitude i = 28 mag, the median uncertainty is $\delta {z}_{\mathrm{photo}}\sim 0.5$, and it reaches $\delta {z}_{\mathrm{photo}}\sim 1$ at $i\approx 30$ mag (Coe et al. 2006). The compilation of Skelton et al. (2014) provides even more robust photometric redshifts, thanks to the expanded photometric data set. The agreement with available spectroscopic redshifts is typically very good in these cases, with a standard deviation on ${\rm{\Delta }}z/(1+z)$ of $\approx 0.01$ (Skelton et al. 2014). In addition, the 3D-HST survey provides HST ACS and WFC3 grism observations of the field, yielding grism redshifts for tens of sources in our pointing (Momcheva et al. 2016). Slit spectroscopy for 74 (mostly bright) galaxies in the field is also available (Le Fèvre et al. 2005; Skelton et al. 2014; Morris et al. 2015). Finally, integral field spectroscopy of this field has been secured with the ESO Very Large Telescope/MUSE. These data are part of a Guaranteed Time observing program targeting the UDF. In particular, a single (1 arcmin²) deep (21 hr on source) pointing overlaps with ∼70% of the ASPECS coverage. The cubes have been processed and analyzed with the improved MUSE GTO pipeline. These observations will be presented in R. Bacon et al. (2016, in preparation).

Within a radius of $34^{\prime\prime}$ from the ALMA 3 mm pointing center (approximately the size of the primary beam of our 3 mm observations), there are 1302 galaxies from the combination of all the available photometric catalogs. We use the high-z extension of MAGPHYS (da Cunha et al. 2008, 2015) to fit the SEDs of all of them. The input photometry includes 26 broad and medium filters ranging from observed U band to Spitzer IRAC 8 μm. Additionally, we include the ASPECS 1 mm continuum photometry for those sources where $\gt 2$-σ emission is reported in our 1 mm continuum image. We do not include any Spitzer MIPS or Herschel PACS photometry because the angular resolutions of these instruments are not sufficient to accurately pinpoint the emission.²⁸ Our MAGPHYS analysis provides us with a posterior probability distribution of the stellar mass (${M}_{* }$), the SFR, the specific SFR (sSFR = SFR/${M}_{* }$), the dust mass (${M}_{\mathrm{dust}}$), and the IR luminosity for each galaxy in the field. We take the 14% and 86% quartiles of the posterior distributions as the uncertainties in the parameters, and we account for an additional fiducial 10% uncertainty that is due to systematics (subtleties in the photometric analysis adopted in the input catalogs, such as aperture corrections and deblending assumptions; zero-point uncertainties; etc.). Figure 1 shows the SFR as a function of M_* for all the 1302 galaxies in our field.

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ID.1 (Table 1, Figures 2 and 11) is a compact galaxy at $z\approx 2.5$. Momcheva et al. (2016) report a grism redshift z = 2.561, based on the detection of the [O ii] line in the 3D-HST data. This redshift is improved by our blind detection of three CO transitions (ASPECS 3 mm.1, 1 mm.1, 1 mm.2; see Paper I), clearly pininng down the redshift to z = 2.543. The HST images show a blue component in the north and a red component in the south (or possibly a single relatively blue component that is partially reddened in the south by a thick dust lane). A group of bright galaxies is present a few arcsec north of this galaxy, but their spectroscopic redshifts show that the group is in the foreground, with only one other source lying at $z\sim 2.5$ (the galaxy $\sim 2^{\prime\prime}$ west of ID.1). The starlight emission coincident with the CO detection is compact, with a scale radius ${R}_{e}\approx 1.7\,\mathrm{kpc}$ (van der Wel et al. 2012). Chandra reveals X-ray emission associated with this galaxy. The measured X-ray flux is ${F}_{{\rm{X}}}=5.7\times {10}^{-17}$ erg s⁻¹ cm⁻², yielding an X-ray luminosity ${L}_{{\rm{X}}}=3.0\times {10}^{42}$ erg s⁻¹ (Xue et al. 2011).

ID.2 (Figure 11) has an HST morphology consistent with a large disk galaxy at z = 1.552. Its slit redshift (z = 1.552, Kurk et al. 2013) matches our CO line detection well (ASPECS 3 mm.2), assuming CO(2–1). The disk has an inclination of $\sim 60^\circ$ (based on the aspect ratio, van der Wel et al. 2012), with an effective radius of 8.3 kpc. The galaxy is detected with Chandra. Xue et al. (2011) report a flux of ${F}_{{\rm{X}}}=3.6\times {10}^{-15}$ erg s⁻¹ cm⁻² (but $2.6\times {10}^{-15}$ erg s⁻¹ cm⁻² in Lehmer et al. 2005), yielding an X-ray luminosity ${L}_{{\rm{X}}}=5.5\times {10}^{43}$ erg s⁻¹, suggesting that ID.2 hosts an active galactic nucleus (AGN).

ID.3 and ID.4 (Figure 12) are the two components of an apparent pair of overlapping spiral galaxies. The southern component exibits bright [O ii] emission at ∼7784 Å (see Figure 3), clearly placing it at z = 1.088 (in agreement with the CO redshift of ASPECS 3 mm.5); the northern component shows bright CO emission (ASPECS 3 mm.3), which could be interpreted as CO(2–1) at z = 1.382. Our careful analysis of the MUSE data around 8880 Å reveals faint [O ii] emission (although contaminated by sky line emission), supporting the CO identification (see Figure 3). The disk of ID.4 has a scale radius of 5.8 kpc based on HST imaging (van der Wel et al. 2012); for ID.3, the estimated radius is 8.3 kpc (but the overlap with the southern component may partially affect this estimate). ID.4 appears as an upper limit in the X-ray catalog by Xue et al. (2011) (${F}_{{\rm{X}}}\lt 6.7\times {10}^{-17}$ erg s⁻¹ cm⁻², ${L}_{{\rm{X}}}\lt 4.3\times {10}^{41}$ erg s⁻¹).

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ID.5 (Figure 13) and ID.8 (Figure 14) lie in a crowded region of our field. Skelton et al. (2014) report a spectroscopic redshift z = 1.047 for ID.5. However, the inspection of the MUSE data reveals two clearly distinguished line sets of the [O ii] doublet at z = 1.038 and z = 1.098. The latter matches the redshift of two CO lines that are slightly too faint to be selected in our blind search (S/N ≈ 4.8, see Paper I). ID.8, on the other hand, is found at another redshift (z = 0.999). No CO emission is found at this position and frequency, although the lowest-J transition that we encompass is CO(4–3) at 1 mm. ID.8 is detected in the X-rays (Xue et al. 2011). Its faintness (${F}_{{\rm{X}}}=8.2\times {10}^{-17}$ erg s⁻¹ cm⁻², ${L}_{{\rm{X}}}=4.3\times {10}^{41}$ erg s⁻¹) seems consistent with a starburst rather than an AGN (Ranalli et al. 2003).

ID.6 (Figure 13) is located $\sim 4^{\prime\prime}$ east of ID.3 and probably belongs to a common physical structure (together with other galaxies with a spectroscopic $z\approx 1.09$). It is detected in the 1 mm continuum, and its CO spectrum shows an ∼3σ excess at the frequency of the expected CO(2–1) line. The CO(4–3) transition is also detected with similar significance, although the best Gaussian fit of the line suggests a velocity shift of ∼200 km s⁻¹ between the two transitions. This is most likely due to the poor S/N of the two lines.

ID.7 (Figure 14) appears as a very compact source (${R}_{{\rm{e}}}=0.7$ kpc) at z = 1.221. Its Chandra image reveals the presence of a bright AGN (${F}_{{\rm{X}}}=1.01\times {10}^{-14}$ erg s⁻¹ cm⁻² in Lehmer et al. (2005); $8.3\times {10}^{-15}$ erg s⁻¹ cm⁻² in Evans et al. (2010); $6.3\times {10}^{-15}$ erg s⁻¹ cm⁻² in Xue et al. 2011), yielding an X-ray luminosity of ${L}_{{\rm{X}}}=5.4\times {10}^{43}$ erg s⁻¹). It is not detected in the 1 mm dust continuum. A 3σ excess is measured at the expected frequency of the CO(2–1) transition.

ID.9 and ID.10 (Figure 15) are both at $z\sim 2.3$. They are among the faintest galaxies in our sample in terms of ${L}_{\mathrm{IR}}$, just above the 10¹¹ ${L}_{\odot }$ cut. ID.9 appears as a compact bulge. ID.10 appears as a spiral galaxy with disturbed morphology. ASPECS data cover three CO transitions in these galaxies: CO(3–2), CO(7–6), and CO(8–7). None of these lines are detected.

ID.11 (Figure 16) is a compact (${R}_{{\rm{e}}}$ = 1.2 kpc) galaxy at z = 0.895. As for ID.8, the lowest-J CO transition in the ASPECS coverage is the CO(4–3), which remains undetected.

We measure the line fluxes (or place limits) for all the CO transitions covered in both the 3 and 1 mm line scans. We extract the CO spectra at the position of the optical coordinates of the sources in our sample. We fit the lowest-J transitions accessible with ASPECS data with a Gaussian profile; in the case of a detection, we fit the higher-J lines imposing the same line width. We consider as detections those cases where the flux obtained in the Gaussian fit is >3× its uncertainty. If the line is not detected, we assume a fiducial line width of 300 km s⁻¹ , and we use the upper boundary of the 3σ confidence range on the flux as upper limit. Table 2 reports the CO line fluxes, shifts compared with the nominal redshift, and the line width. The detected sources in our sample have a median CO flux of 0.19 Jy km s⁻¹ (considering only the lowest-J transition observed in each object). For a comparison, the detected main-sequence galaxies in Tacconi et al. (2013) have a median CO flux of 0.57 Jy km s⁻¹, i.e., $3\times$ higher than the median flux of our detections.

Table 2.  CO Lines in the Galaxies of our Sample

ID	${J}_{\mathrm{up}}$	${\rm{\Delta }}v$	${F}_{\mathrm{line}}$	FWHM
		(km s−1)	(Jy km s−1)	(km s−1)
(1)	(2)	(3)	(4)	(5)
1	3	−45 ± 8	${0.723}_{-0.003}^{+0.003}$	504 ± 12
1	7	−150 ± 120	${0.786}_{-0.006}^{+0.006}$	504a
1	8	−45 ± 70	${1.098}_{-0.005}^{+0.005}$	504a
2	2	135 ± 9	${0.443}_{-0.007}^{+0.007}$	538 ± 13
2	5	135 ± 45	${0.502}_{-0.090}^{+0.090}$	538a
2	6	−45 ± 45	${0.820}_{-0.100}^{+0.100}$	538a
3	2	−37 ± 8	${0.135}_{-0.003}^{+0.003}$	57 ± 12
3	5	⋯	$\lt 0.021$	57a
4	2	52 ± 7	${0.180}_{-0.006}^{+0.006}$	82 ± 11
4	4	⋯	$\lt 0.121$	82a
5	2	220 ± 35	${0.190}_{-0.040}^{+0.040}$	352 ± 11
5	4	−28 ± 40	${0.390}_{-0.065}^{+0.065}$	352a
6	2	−160 ± 70	${0.340}_{-0.070}^{+0.060}$	530 ± 11
6	4	230 ± 70	${0.370}_{-0.090}^{+0.090}$	182a
7	2	150 ± 17	${0.104}_{-0.029}^{+0.019}$	150 ± 11
7	5	⋯	$\lt 0.106$	150a
8	4	⋯	$\lt 0.059$	⋯
9	3	⋯	$\lt 0.076$	⋯
9	7	⋯	$\lt 0.012$	⋯
9	8	⋯	$\lt 0.230$	⋯
10	3	⋯	$\lt 0.048$	⋯
10	6	⋯	$\lt 0.144$	⋯
10	7	⋯	$\lt 0.465$	⋯
11	4	⋯	$\lt 0.015$	⋯

Note. (1) Source ID. (2) Upper J of the CO transition. (3) Velocity shift, compared with the redshift quoted in Table 1. (4) Line flux. (5) Line width, expressed as the full width at half maximum (FWHM) from the Gaussian fit.

^aFixed from the fit of a lower J line.

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The luminosity of the lowest-J transitions observed in our molecular scans is transformed into the equivalent ground-state luminosity ${L}_{\mathrm{CO}(1-0)}^{\prime }$ using ${L}_{\mathrm{CO}(J-[J-1])}^{\prime }/{r}_{J1}$, where we adopt the recent CO excitation ladder of main-sequence galaxies derived by Daddi et al. (2015): r₂₁ = 0.76 ± 0.09, ${r}_{31}=0.42\pm 0.07$, and ${r}_{41}=0.31\pm 0.06$.³⁰ The uncertainty in ${L}_{\mathrm{CO}(1-0)}^{\prime }$ accounts for both the measured flux uncertainty and the standard deviation in the r_J1 values in the sample studied by Daddi et al. (2015). The molecular gas masses are then derived as

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{{\rm{H}}2}}{{M}_{\odot }}=\displaystyle \frac{{\alpha }_{\mathrm{CO}}}{{r}_{J1}}\,\displaystyle \frac{{L}_{(J-[J-1])}^{\prime }}{{\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2}}.\end{eqnarray} \tag{ 1 }$$

We adopt ${\alpha }_{\mathrm{CO}}=3.6$ ${M}_{\odot }$ (K km s⁻¹ pc²)⁻¹ (Daddi et al. 2010b). This conversion factor has been demonstrated to be appropriate for main-sequence galaxies through comparisons with dynamical masses (Daddi et al. 2010b), CO line SED-fitting (Daddi et al. 2015), and detailed dust-SED modeling (Genzel et al. 2015). In Section 5.5 we further discuss the implications of our ${\alpha }_{\mathrm{CO}}$ assumption. Table 3 lists the values of molecular gas masses that we derive for each source. We then combine these measurements or limits on the molecular gas mass with properties of the galaxies inferred from the SED fitting (in particular, the stellar mass M_* and the SFR) to compute the molecular-to-stellar mass ratio ${M}_{{\rm{H}}2}/{M}_{* }$ and the depletion timescale ${t}_{\mathrm{dep}}={M}_{{\rm{H}}2}/\mathrm{SFR}$ (see Table 3).

Table 3.  CO Luminosities and CO-based Galaxy Parameters

ID	z	${J}_{\mathrm{up}}$	$L^{\prime}$	${L}_{\mathrm{CO}(1-0)}^{\prime }$	${M}_{{\rm{H}}2}$	${M}_{{\rm{H}}2}/{M}_{* }$	${t}_{\mathrm{depl}}$
			$(\times {10}^{9}$ K km s−1 pc2)	$(\times {10}^{9}$ K km s−1 pc2)	$(\times {10}^{9}$ ${M}_{\odot }$)		(Gyr)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
1	2.543	3	${24.03}_{-0.10}^{+0.10}$	${57}_{-10}^{+10}$	${206}_{-34}^{+34}$	${12}_{-2}^{+2}$	${3.3}_{-0.6}^{+0.7}$
2	1.551	2	${13.71}_{-0.27}^{+0.21}$	${18}_{-2}^{+2}$	${65}_{-8}^{+8}$	${0.24}_{-0.05}^{+0.05}$	${0.9}_{-0.4}^{+0.6}$
3	1.382	2	${3.364}_{-0.08}^{+0.07}$	${4.4}_{-0.5}^{+0.5}$	${15.9}_{-1.9}^{+1.9}$	${0.30}_{-0.07}^{+0.08}$	${0.9}_{-0.3}^{+0.6}$
4	1.088	2	${2.831}_{-0.09}^{+0.09}$	${3.7}_{-0.5}^{+0.5}$	${13.4}_{-1.7}^{+1.7}$	${0.48}_{-0.11}^{+0.13}$	${0.6}_{-0.3}^{+0.4}$
5	1.098	2	${3.089}_{-0.66}^{+0.70}$	${4.1}_{-1.0}^{+1.0}$	${15}_{-4}^{+4}$	${2.5}_{-0.6}^{+0.7}$	${0.33}_{-0.09}^{+0.09}$
6	1.094	2	${5.388}_{-1.16}^{+0.91}$	${7.1}_{-1.7}^{+1.5}$	${25}_{-6}^{+5}$	${0.34}_{-0.09}^{+0.10}$	${1.6}_{-0.7}^{+0.9}$
7	1.221	2	${2.047}_{-0.57}^{+0.37}$	${2.7}_{-0.8}^{+0.6}$	${10}_{-3}^{+2}$	${0.6}_{-0.2}^{+0.16}$	${0.066}_{-0.020}^{+0.016}$
8	0.999	4	$\lt 0.20$	$\lt 0.63$	$\lt 2.3$	$\lt 0.03$	$\lt 0.06$
9	2.447	3	$\lt 2.4$	$\lt 5.6$	$\lt 21$	$\lt 8$	$\lt 1.8$
10	2.224	3	$\lt 2.2$	$\lt 5.3$	$\lt 19$	$\lt 1.6$	$\lt 0.9$
11	0.895	4	$\lt 0.53$	$\lt 1.7$	$\lt 6.2$	$\lt 0.4$	$\lt 0.15$

Notes. (1) Source ID. (2) Redshift. (3) Observed transition. (4) Line luminosity. (5) Equivalent CO(1–0) luminosity, assuming the ${r}_{J1}$ ratios in Daddi et al. (2015). (6) Molecular gas mass, assuming ${\alpha }_{\mathrm{CO}}=3.6$ ${M}_{\odot }$ (K km s⁻¹ pc²)⁻¹. (7) Molecular-to-stellar mass ratio, ${M}_{{\rm{H}}2}/{M}_{* }$. (8) Depletion time, ${t}_{\mathrm{dep}}={M}_{{\rm{H}}2}/\mathrm{SFR}$.

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In the case of ID.2, our ALMA observations spatially resolve the CO(2–1) emission over $\gt 15\,\mathrm{kpc}$, despite the relatively coarse spatial resolution of the 3 mm data. A clear velocity gradient is observed in the line emission, as shown in Figure 4. While the resolution and the signal-to-noise ratio are too poor for an accurate modeling of the gas dynamics, we obtain an estimate of the dynamical mass assuming that the gas is rotating in a disk with the inclination derived from the HST near-IR imaging (PA = −55°, inclination = 60° with respect to the line of sight). We then assume a radial distribution of the mass that scales as ${M}_{\mathrm{dyn}}\propto {R}^{\gamma }$, where γ = 1 yields the flat rotation curves typically observed in galaxies; γ = 2 implies a constant surface density of mass in the disk, and yields ${v}_{\mathrm{rot}}\propto \sqrt{R};$ and γ = 3 corresponds to a solid rotator (${v}_{\mathrm{rot}}\propto R$). We then generated mock velocity maps for these three cases, assuming that the CO light traces the mass distribution; and we inferred expected line profiles (see Figure 4). The γ = 1 case shows the typical "double-horned" profile observed in local spiral galaxies. This seems to provide a better description of the observed CO(2–1) line than the other two models, which fail to reproduce the extension of the blue wing of the line. The implied dynamical mass is ${M}_{\mathrm{dyn}}\approx 2\times {10}^{11}$ ${M}_{\odot }$ at R = 8.3 kpc (= the effective radius). We stress, however, that this estimate is highly dependent on the model assumptions. A firmer estimate of the dynamical mass in this galaxy requires deeper data at higher spatial resolution.

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ID.2 also appears in the SINFONI Integral field spectroscopy survey in the near-IR (SINS; Förster Schreiber et al. 2009) as GMASS-1084 (see also Kurk et al. 2013). SINS investigated the morphology and kinematics of ionized gas (as traced by the Hα hydrogen line) in a sample of galaxies at $z=1\mbox{--}3.5$. The Hα line in ID.2 is emitted on a smaller region (half-light radius ${R}_{1/2}=3.1\pm 1.0$ kpc) than the CO. The observed Hα circular velocity is 67 ± 9 km s⁻¹, which is corrected into 230 km s⁻¹ by assuming a low-inclination angle ($\sim 20^\circ$). This yields a dynamical mass of $1.2\times {10}^{11}$ ${M}_{\odot }$, roughly consistent with our estimate, especially if one considers that the high level of dust reddening (A_V = 2.4 mag from our global MAGPHYS fit) in this source may be responsible for suppressing Hα in parts of this galaxy. We note, however, that the SED fit of this source in Förster Schreiber et al. (2009) yields a stellar mass of only ${M}_{* }={3.61}_{-0.60}^{+0.34}\times {10}^{10}$ ${M}_{\odot }$ and a high SFR = ${490}_{-31}^{+190}$ ${M}_{\odot }$ yr⁻¹ (i.e., ${L}_{\mathrm{IR}}\approx 5.7\times {10}^{12}$ ${L}_{\odot }$). This last estimate disagrees with our dust continuum measurements: e.g., assuming a modified blackbody template with β = 1.6 and ${T}_{\mathrm{dust}}=25$ K, such a high SFR would imply a dust-continuum flux density of 11 mJy at 1.2 mm (observed: 0.22 ± 0.02 mJy) and of 32 mJy at 160 μm (observed: 6.9 ± 0.3 mJy).

As seen from Figure 4, the molecular gas, as traced through CO emission, is extended on scales of $\gt 15\,\mathrm{kpc}$ ($\gt 2^{\prime\prime}$ at z = 1.552), i.e., comparable to that of the stellar disk. On the other hand, the 1 mm dust continuum is unresolved at $1.5^{\prime\prime} \times 1.0^{\prime\prime}$ resolution, i.e., it is significantly smaller than that of the CO (see Figure 11). This is not an effect of interferometric filtering or sensitivity of the 1 mm data. When we convolve the 1 mm continuum data to the synthesized beam of the 3 mm data, we do not recover the size seen in CO emission. This serves as a cautionary note that CO and dust sizes may not be the same. As a consequence, the masses deduced from these measurements may trace different regions or components in the galaxy (for other examples of a mismatch between CO and dust morphology in high-redshift galaxies, see Riechers et al. 2011; Hodge et al. 2015; Spilker et al. 2015). This may explain some of the differences between gas mass estimates derived from CO and dust imaging, with the gas masses derived from dust emission being typically lower than those derived from CO (see Section 5.5): At the observed wavelength (1.2 mm), dust is optically thin (with the only exception of ID.1, all the sources in our sample globally have ${{\rm{\Sigma }}}_{\mathrm{gas}}\ll {10}^{4}$ ${M}_{\odot }$ pc⁻², i.e., ${N}_{{\rm{H}}2}\ll {10}^{24}$ cm⁻²; this yields τ[242 GHz]$\,\ll \,0.1$ for solar metallicities, adopting the Draine & Lee 1984 formalism). The CO low-J emission, on the other hand, is optically thick practically everywhere in galaxies.

As shown in Table 1, ASPECS covers 2–3 different CO transitions in 9 out of 11 galaxies in our sample. Figure 5 shows the inferred constraints on the CO excitation ladder. In ID.1, all three observed transitions [CO(3–2), CO(7–6), and CO(8–7)] were detected in our blind search for line emission (ASPECS 3 mm.1, 1 mm.1, and 1 mm.2, respectively). In ID.2, the CO(2–1) line appears in the results of our blind search (ASPECS 3 mm.2). The CO(5–4) and CO(6–5) lines are also observed, but because of their lower significance, they were not detected in our blind search. In particular, the CO(6–5) line is very noisy, as it is found at the high-frequency end of the 1 mm spectral scan, and it is spatially located at the edge of our mosaic. The CO(2–1) transitions in ID.3 and ID.4 are also identified in our blind search (ASPECS 3 mm.3 and 3 mm.5, respectively). However, the CO(5–4) line in ID.3 and the CO(4–3) line in ID.4 are not detected. In particular for ID.3, this places very strong limits on the CO excitation of this galaxy, significantly below the average CO ladder of the Milky Way disk (see Figure 5). In ID.5 and ID.6, we detect both CO(2–1) and CO(4–3). Finally, in ID.7 we only have a tentative detection of CO(2–1), while the CO(5–4) transition remains undetected. No other line is detected in the remainder of our sample.

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In Figure 5 we compare our measurements and limits with the CO excitation templates of the Milky Way disk, and of the starburst in M82 (Weiß et al. 2007). Additionally, we compare with the average template for high-z main-sequence galaxies by Daddi et al. (2015) and with the theoretical predictions based on the SFR surface density by Narayanan & Krumholz (2014) (see Table 1 and the discussion in Section 5.2.2). In no case do we find starburst-like CO excitation that would be comparable with the center of M82 (Weiß et al. 2007) or with what is typically observed in high-z sub-mm galaxies (SMGs; e.g., Bothwell et al. 2013; Spilker et al. 2014). ID.1 shows a CO(7–6)/CO(3–2) ratio r₇₃ = 0.2, consistent with a high-density photon-dominated region (Meijerink et al. 2007). On the other hand, the CO(8–7) transition appears brighter, implying that a high-excitation component of the interstellar medium (ISM) might be in place. Interestingly, ID.2 shows a lower excitation (in particular in the CO[5–4]/CO[2–1] ratio, which is consistent with Milky Way excitation). This difference in CO excitation is remarkable when we consider that ID.1 is not detected with Chandra (${L}_{{\rm{X}}}\lt 6\times {10}^{42}$ erg s⁻¹), whereas ID.2 shows a bright X-ray detection, indicative of the presence of a central AGN. The X-ray emission from the AGN can boost the emission of high-J CO transitions. The CO(7–6)/CO(3–2) ratio r₇₃ is typically 0.16–0.63 in high-density photon-dominated regions that are powered by star formation (as in ID.2), but it can reach values as high as r₇₃ = 30 in the presence of intense X-ray illumination (Meijerink et al. 2007). This might explain the higher CO excitation observed in high-z QSOs with respect to submillimeter galaxies (Carilli & Walter 2013). The lack of such high-excitation feature suggests that the central AGN activity in ID.2 has no major impact on its global CO properties. We attribute the higher excitation in ID.1 to the much more compact emission in this galaxy. As shown in Table 1, ID.1 has a radius that is only ∼1/5 of that of ID.2, which translates into a difference in surface area of ∼24. Our MAGPHYS-based SFR estimates are comparable (∼70 ${M}_{\odot }$ yr⁻¹), thus the surface density of star formation (${{\rm{\Sigma }}}_{\mathrm{SFR}}$) is much higher in ID.1. This is also discussed in Section 5.2 below. The increased radiation field intensity caused by the high SFR surface density and/or the higher gas density are very likely the reason for the increased CO excitation (see Narayanan & Krumholz 2014).

The stellar masses of the galaxies in our sample range between $(2.8\mbox{--}275)\times {10}^{9}$ ${M}_{\odot }$ (two orders of magnitude). The ${L}_{\mathrm{IR}}\gt {10}^{11}$ ${L}_{\odot }$ cut in our sample definition selects sources with SFR>10 ${M}_{\odot }$ yr⁻¹. The measured SFRs range between 12–150 ${M}_{\odot }$ yr⁻¹.³¹

Figure 1 shows the location of our galaxies in the ${M}_{* }$–SFR ("main sequence") plane. We plot all the galaxies in the field with an F850LP or F160W magnitude brighter than 27.5 mag (this cut allows us to remove sources with highly uncertain SED fits). The galaxies in the present sample are highlighted with large symbols. The different redshifts of the sources are indicated by different colors. As expected from the known evolution of the "main sequence" of star-forming galaxies (e.g., Whitaker et al. 2012; Schreiber et al. 2015), sources at higher redshifts tend to have a higher SFR per unit stellar mass. Comparing with the Herschel-based results by Schreiber et al. (2015), we find that half of the galaxies in our sample (ID.1, 2, 4, 8, 9, 10) lie on the main sequence (within a factor $3\times$) at their redshift. Three galaxies (ID.5, 7, 11) are above the main sequence (in the "starburst" region), and the remaining two galaxies (ID.3 and ID.6) show an SFR $\sim \,3\times$ lower than main-sequence galaxies at these redshifts and stellar masses. Similar conclusions are reached when we compare our results with the main-sequence fits by Whitaker et al. (2012) (see Figure 1).

The relationship between the total infrared luminosity (${L}_{\mathrm{IR}}$, a proxy for the SFR) and the total CO luminosity (${L}_{\mathrm{CO}}^{\prime }$, a proxy for the available gas mass) of galaxy samples is typically referred to as the "integrated Schmidt–Kennicutt" law (Schmidt 1959; Kennicutt 1998; Kennicutt & Evans 2012), or, more generally, the "star formation" law. Sometimes average surface density values are derived from these quantities, resulting in average surface SFR densities (${{\rm{\Sigma }}}_{\mathrm{SFR}}$) and gas densities (${{\rm{\Sigma }}}_{\mathrm{gas}}$). We here explore both relations.

5.2.1. Global Star Formation Law: IR Versus CO Luminosities

In Figure 6 we compare the IR and CO(1–0) luminosities of our sources with respect to a compilation of galaxies both at low and high redshift from the review by Carilli & Walter (2013), and with the secure blind detections in Decarli et al. (2014). For galaxies in our sample that are undetected in CO, we plot the corresponding 3σ limit on the line luminosities. The IR–CO luminosity empirical relation motivates the ${L}_{\mathrm{IR}}$ cut in our sample selection, as galaxies with ${L}_{\mathrm{IR}}\gt {10}^{11}$ ${L}_{\odot }$ should have CO emission brighter than $L^{\prime} \approx 3\times {10}^{9}$ K km s⁻¹ pc² (i.e., our typical sensitivity limit in ASPECS; see Paper I). All the galaxies in our sample should therefore be detected in CO. We find that most of the CO-detected galaxies in our sample lie along the one-to-one relation, followed by local spiral galaxies as well as color-selected main-sequence galaxies at $1\lt z\lt 3$ (Daddi et al. 2010b; Genzel et al. 2010, 2015; Tacconi et al. 2013). Only two galaxies significantly deviate: ID.5, which appears on the upper envelope of the IR–CO relation, close to high-redshift starburst galaxies; and ID.7, which is largely underluminous in CO for its bright IR emission. As discussed in the previous section, these two galaxies appear as starbursts in Figure 1. Moreover, ID.7 hosts a bright AGN. If the AGN contamination at optical wavelengths is significant, then our MAGPHYS-based SFR estimate is likely in excess (since MAGPHYS would associate some of the AGN light at rest-frame optical and UV wavelengths to a young stellar population), thus explaining the large vertical offset of this galaxy with respect to the "star formation law" shown in Figure 6. Notably, out of the 4 CO non-detections in our sample, ID.9 and ID.10 are still consistent with the relation, while ID.8 and ID.11 are not. These two galaxies are located at z = 0.999 and z = 0.895, respectively. The lowest-J transition sampled in our study is CO(4–3). Their non-detections might be explained if the excitation in these two sources were much lower than what we assumed to infer ${L}_{\mathrm{CO}(1-0)}^{\prime }$ (${r}_{41}=0.31;$ see Section 4.1).

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The sources that are also detected in the blind search for CO (ID.1, 2, 3, 4) tend to lie on the lower "envelope" of the plot. This is expected, as these galaxies have been selected based on their CO luminosity (x-axis).

Figure 6 also shows the x-axis position of the remaining CO blind detections from the 3 mm search in Paper I. The CO luminosities of these lines are uncertain (the line identification is ambiguous in many cases, and a fraction of these lines is expected to be a false positive; see Paper I); however, it is interesting to note that these sources typically populate ranges of line luminosities that were previously unexplored at $z\gt 1$ (see similar examples in Chapman et al. 2008, 2015b; Casey et al. 2011), and comparable with or even lower than the typical dust luminosities of local spiral galaxies. We emphasize that a significant fraction of these lines is expected to be real (see Paper I). Deeper data are required to better characterize these candidates.

5.2.2. Average Surface Densities: SFR Versus Gas Mass

We infer average estimates of ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ and ${{\rm{\Sigma }}}_{\mathrm{gas}}$ by dividing the global SFR and ${M}_{{\rm{H}}2}$ of the galaxy by a fiducial area set by the size of the stellar component, as CO and optical radii are typically comparable (Schruba et al. 2011; Tacconi et al. 2013). We thus use the information from the stellar morphology derived by van der Wel et al. (2012) and reported in Table 1 to infer ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ = SFR/($2\,\pi \,{R}_{e}^{2}$) and ${{\rm{\Sigma }}}_{{\rm{H}}2}$ = ${M}_{{\rm{H}}2}$/($2\,\pi \,{R}_{e}^{2}$), where ${M}_{{\rm{H}}2}$ is our CO-based measurement of the molecular gas mass, and the factor 2 is due to the fact that the R_e includes only half of the light of the galaxy (see a similar approach in Tacconi et al. 2013).

In Figure 7 we show the star formation law for average surface densities. Global measurements of local spiral galaxies and starbursts are taken from Kennicutt (1998), and corrected for the updated SFR calibration following Kennicutt & Evans (2012) and to the ${\alpha }_{\mathrm{CO}}$ value adopted in this paper. We also plot the galaxies in the IRAM Plateau de Bure HIgh-z Blue Sequence Survey (PHIBSS; Tacconi et al. 2013), again corrected to match the same ${\alpha }_{\mathrm{CO}}$ assumption used in this work, and the secure detections in Decarli et al. (2014). Interestingly, the two CO-brightest galaxies in our sample, ID.1 and ID.2, appear to populate opposite extremes of the density ranges observed in high-z galaxies: ID.1 appears very compact, thus reaching the top right corner of the plot (${{\rm{\Sigma }}}_{\mathrm{gas}}\approx {\rm{10,000}}$ ${M}_{\odot }$ pc⁻²). On the other hand, in ID.2 the vast gas reservoir is spread over a large area (as apparent in Figure 4), thus yielding a globally low ${{\rm{\Sigma }}}_{\mathrm{gas}}$. We also find that most of the sources in our sample lie along the ${t}_{\mathrm{depl}}\approx 1\,\mathrm{Gyr}$ line, in agreement with local spiral galaxies and the PHIBSS main-sequence galaxies. Only ID.7 and ID.8 lie closer to the ${t}_{\mathrm{depl}}\approx 0.1\,\mathrm{Gyr}$ line. In particular, the offset of ID.7 with respect to the bulk of the sample in the context of the global star formation law (Figure 6) is combined here with the very compact size of the emitting region, thus isolating the source in the top left corner of the plot (see Figure 7). Once again, a significant AGN contamination in the estimates of both the rest-frame optical/UV luminosity and in the size of the emitting region could explain such an outlier. We also caution that, in some of these galaxies, optical and CO radii might differ.

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Figure 8 shows the depletion time, ${t}_{\mathrm{depl}}={M}_{{\rm{H}}2}/\mathrm{SFR}$, as a function of the specific SFR. This timescale sets how quickly the gas is depleted in a galaxy given the currently observed SFR (ignoring any gas repleneshing). Our data are compared again with the secure blind detections in Decarli et al. (2014), with the PHIBSS sample, and with the sample of starburst galaxies studied by Silverman et al. (2015) (in the latter case, we do not change the adopted value of ${\alpha }_{\mathrm{CO}}=1.1$ ${M}_{\odot }$(K km s⁻¹ pc²)⁻¹, as these are not main-sequence galaxies). Starburst galaxies tend to reside in the bottom right corner of the plot (they are highly star forming given their stellar mass, and they are using up their gaseous reservoir fast). Galaxies with large gas reservoirs and mild star formation populate the top left corner of the plot. Since the IR luminosity is proportional to the SFR and the CO luminosity is used to infer ${M}_{{\rm{H}}2}$, the y-axis of this plot conceptually corresponds to a diagonal line (top left to bottom right) in Figure 6. In addition, the diagonal lines in Figure 8 mark the loci of the constant molecular-to-stellar mass ratio ${M}_{{\rm{H}}2}/{M}_{* }$.

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The sources in our sample range over almost 2 dex in sSFR and ${t}_{\mathrm{depl}}$. Noticeably, ID.1 is highly star forming (it resides slightly above the main sequence of star-forming galaxies at $z\sim 2.5$, see Figure 1), so we would expect it to reside in the bottom right corner of Figure 8; however, its gaseous reservoir is very large for its IR luminosity (see also Figure 6), thus placing ID.1 in the top right corner of the plot (${M}_{{\rm{H}}2}/{M}_{* }=12$). On the other hand, ID.2 hosts an enormous reservoir of molecular gas, but because of its even higher stellar mass (yielding low sSFR), it resides on the left side of the plot (${M}_{{\rm{H}}2}/{M}_{* }=0.24$). Their depletion timescales, however, are comparable (1–3 Gyr). We stress that these results are based on very high S/N CO line detections, and on very solid descriptions of the galaxy SEDs (see Figure 11). The sources that populate the starburst region in Figure 1 and reside in the top left part of Figure 6 (in particular, ID.5 and ID.7) consistently appear in the bottom right corner of Figure 8, that is, among starbursts.

A useful parameter to investigate the molecular gas content in high-z galaxies is the molecular gas to stellar mass ratio, ${M}_{{\rm{H}}2}/{M}_{* }$. We prefer this parameter rather than the molecular gas fraction, ${f}_{\mathrm{gas}}={M}_{{\rm{H}}2}/({M}_{* }+{M}_{{\rm{H}}2})$, as the two involved quantities (${M}_{{\rm{H}}2}$ and M_*) appear independently at the numerator and denominator of the fraction, so that the parameter is well defined even when we only have upper limits on ${M}_{{\rm{H}}2}$. Figure 9 shows the dependence of ${M}_{{\rm{H}}2}/{M}_{* }$ on redshift in the galaxies of our samples and in galaxies from the literature. This plot informs us of the typical gas content as a function of cosmic time, and can help us shed light on the origin of the cosmic star formation history (see, e.g., Geach et al. 2011; Magdis et al. 2012, and Paper III of this series). Color-selected star-forming galaxies close to the epoch of galaxy assembly are claimed to show large ${M}_{{\rm{H}}2}/{M}_{* }$, with reservoirs of gas as large as (or even larger than) the stellar mass (i.e., ${M}_{{\rm{H}}2}/{M}_{* }\sim 1;$ see, e.g., Daddi et al. 2010a; Tacconi et al. 2010, 2013). Indeed, we find examples of very high gas fractions: ID.1 (${M}_{{\rm{H}}2}/{M}_{* }=12$) and the starburst galaxy ID.5 (${M}_{{\rm{H}}2}/{M}_{* }=2.5$) are the most extreme cases. However, it is interesting to note that we also find galaxies with very modest gas fractions, such as ID.2 (${M}_{{\rm{H}}2}/{M}_{* }=0.24$). The CO-detected galaxies at $1.0\lt z\lt 1.7$ in our sample show an average ${M}_{{\rm{H}}2}/{M}_{* }$ ratio that is $\sim 2\times \,$ lower than the average value for the PHIBSS sample at the same redshift, and closer to the global trend established in Geach et al. (2011) and Magdis et al. (2012). The non-detection of CO in ID.8 places particularly strict limits (${M}_{{\rm{H}}2}/{M}_{* }\phantom{\rule{0.25}{0ex}}\lt 0.03$). If the lack of detection is attributed to the very low CO excitation in this galaxy, then it would take a 10× lower r₄₁ (i.e., ${r}_{41}\approx 0.03$) to shift ID.8 on the average trend reported by Geach et al. (2011).

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In addition to the CO line measurements, 6 of the 11 galaxies in our sample also have detections in the 1 mm dust continuum. We can thus estimate the mass of the molecular gas independently of the CO data. The Rayleigh–Jeans part of the dust emission is only weakly dependent on the dust temperature, thus it can be used to trace the mass of dust. Using the dust-to-gas scaling (see, e.g., Sandstrom et al. 2013), it is possible to infer the gas mass via the dust mass.

Groves et al. (2015) compare CO-based gas masses with the monochromatic luminosity of the dust continuum in the Rayleigh–Jeans tail. Their analysis relies on a detailed study of 37 local spiral galaxies in the KINGFISH sample (Kennicutt et al. 2011). The galaxy luminosity in the Herschel/SPIRE 500 μm band is found to scale almost linearly with the gas mass, yielding

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{\mathrm{gas}}}{{10}^{10}\,{M}_{\odot }}=28.5\,\displaystyle \frac{\nu {L}_{\nu }(500\,\mu {\rm{m}})}{{10}^{10}\,{L}_{\odot }}.\end{eqnarray} \tag{ 2 }$$

We compute the rest-frame luminosity $\nu {L}_{\nu }$(500 μm) from the observed 1 mm continuum of the galaxies in our sample. For the k-correction, we adopt a modified blackbody with ${T}_{\mathrm{dust}}$ = 25 K and β = 1.6 (see, e.g., Beelen et al. 2006), shifted at the redshift of each source. Since the observing frequency (242 GHz) falls close to the rest-frame 500 μm (as most of our sources reside at $z\sim 1.2$), and we are sampling the Rayleigh–Jeans tail (which is almost insensitive to the dust temperature), the differences in the corrections due to the adopted templates are negligible for the purposes of this analysis. The adopted values for the k correction, as well as the resulting gas masses, are listed in Table 4.

Table 4.  Gas Mass Estimates Based on the Dust Continuum

ID	z	${F}_{\nu }$(1.2mm)	k-corr	log ${M}_{\mathrm{gas},\mathrm{Groves}}$	log ${M}_{\mathrm{ISM},\mathrm{Scoville}}$	log ${M}_{\mathrm{gas},\mathrm{MAGPHYS}}$
		(μJy)		(${M}_{\odot }$)	(${M}_{\odot }$)	(${M}_{\odot }$)
(1)	(2)	(3)	(4)	(5)	(6)	(7)
1	2.543	552.7 ± 13.8	0.374	${11.02}_{-0.011}^{+0.011}$	${10.69}_{-0.011}^{+0.011}$	${10.53}_{-0.17}^{+0.17}$
2	1.551	223.1 ± 21.6	0.919	${10.63}_{-0.04}^{+0.04}$	${10.33}_{-0.04}^{+0.04}$	${10.09}_{-0.14}^{+0.13}$
4	1.088	96.5 ± 24.7	1.665	${10.24}_{-0.10}^{+0.13}$	${9.95}_{-0.13}^{+0.10}$	${9.78}_{-0.20}^{+0.18}$
5	1.098	46.4 ± 14.9	1.641	${9.93}_{-0.12}^{+0.17}$	${9.63}_{-0.17}^{+0.12}$	${9.25}_{-0.19}^{+0.14}$
6	1.094	69.6 ± 18.9	1.650	${10.10}_{-0.10}^{+0.14}$	${9.81}_{-0.14}^{+0.10}$	${9.58}_{-0.18}^{+0.17}$
10	2.224	36.7 ± 13.8	0.478	${9.86}_{-0.14}^{+0.20}$	${9.53}_{-0.20}^{+0.14}$	${9.25}_{-0.20}^{+0.17}$

Note. Only sources detected at 1 mm in ASPECS are considered. (1) Source ID. (2) Redshift. (3) Observed 242, GHz = 1.2, mm continuum flux density (see Paper II). (4) k correction, expressed as the ratio between the flux density computed at ${\lambda }_{\mathrm{restframe}}=500$ μm and the one at ${\lambda }_{\mathrm{obs}}=1.2\,\mathrm{mm}$, assuming a modified blackbody template for the dust emission with β = 1.6 and ${T}_{\mathrm{dust}}=25$ K. (5) Gas mass based on the 1 mm flux density, derived following Equation (2) (Groves et al. 2015). (6) Gas mass based on the 1 mm flux density, derived following Equation (3) (Scoville et al. 2014, 2016). (7) Gas mass derived from the dust-mass estimate resulting from MAGPHYS SED fitting, assuming a dust-to-gas ratio DGR = 1/100.

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A similar approach was presented by Scoville et al. (2014, 2016). This calibration is tuned on a set of relatively massive $[(0.2\mbox{--}4)\times {10}^{11}$ ${M}_{\odot }$] star-forming galaxies (30 local star-forming galaxies, 12 low-redshift ultraluminous IR galaxies (ULIRGs), and 30 SMGs at z = 1.4–3.0), all having literature observations of the CO(1–0) transition. The tight relation observed between CO(1–0) luminosity and the rest-frame 850 μm monochromatic luminosity (see Figure 1 in Scoville et al. 2016) suggests that a simple conversion factor can be used to derive gas masses from monochromatic dust continuum observations. By setting the dust temperature to ${T}_{\mathrm{dust}}=25$ K (following Scoville et al. 2014), we derive from Equation (12) in their paper ${M}_{\mathrm{ISM}}$ from our 1 mm flux densities as follows:

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{\mathrm{ISM}}}{{10}^{10}\,{M}_{\odot }}=\displaystyle \frac{1.20}{{(1+z)}^{4.8}}\,\displaystyle \frac{{F}_{\nu }}{\mathrm{mJy}}{\left(\displaystyle \frac{\nu }{350\mathrm{GHz}}\right)}^{-3.8}\,\displaystyle \frac{{{\rm{\Gamma }}}_{0}}{{{\rm{\Gamma }}}_{\mathrm{RJ}}}{\left(\displaystyle \frac{{D}_{{\rm{L}}}}{\mathrm{Gpc}}\right)}^{2},\end{eqnarray} \tag{ 3 }$$

where F_ν is the observed dust continuum flux density at the observing frequency ν (242 GHz in our case), ${D}_{{\rm{L}}}$ is the luminosity distance, and ${{\rm{\Gamma }}}_{\mathrm{RJ}}$ is a unitless correction factor that accounts for the deviation from the ${\nu }^{2}$ scaling of the Rayleigh–Jeans tail. In the reference sample of local galaxies, low-redshift ULIRGs, and high-z SMGs that Scoville et al. (2014) used to calibrate Equation (3), ${{\rm{\Gamma }}}_{\mathrm{RJ}}={{\rm{\Gamma }}}_{0}=0.71$. The resulting ISM masses are listed in Table 4.

Finally, we can infer an estimate of ${M}_{\mathrm{gas}}$ from the estimate of the dust mass, ${M}_{\mathrm{dust}}$, that we obtain via our MAGPHYS fit of the available SED, simply scaled by a fixed dust-to-gas mass ratio (DGR). Sandstrom et al. (2013) investigate the dust and gas content in a sample of local spiral galaxies and find DGR ≈ 1/70. Genzel et al. (2015) and Berta et al. (2016) perform a detailed analysis of both gas- and dust-mass estimates in galaxies at $0.9\lt z\lt 3.2$ observed with Herschel, and find a lower value of DGR ≈ 1/100, which is the value we adopt here. We stress that there is a factor $\gt 2\times$ scatter in the estimates of DGR owing to its dependence on M_* and metallicity (Sandstrom et al. 2013; Berta et al. 2016). Following the fundamental metallicity relation in Mannucci et al. (2010), we estimate that galaxies in our sample typically have solar metallicities (the lowest metallicity estimates are for ID.9: Z = 0.6 Z${}_{\odot };$ and ID.5: Z = 0.7 Z${}_{\odot }$), therefore we do not foresee large intra-sample variations of DGR. For simplicity, in our analysis we thus assume a fixed DGR = 1/100. While SED fits are available for all the galaxies in our sample, here we consider only those with a 1 mm detection, in order to best anchor the Rayleigh–Jeans tail of the dust emission. The resulting masses are listed in Table 4.

Figure 10 compares the gas estimates based on Equation (2), following Groves et al. (2015); those obtained via Equation (3), following Scoville et al. (2014); and the estimates based on dust from the MAGPHYS SED fits, with our CO-based estimate (assuming ${\alpha }_{\mathrm{CO}}=3.6$ ${M}_{\odot }$ [K km s⁻¹ pc²]⁻¹). The dust-based gas estimates obtained with different approaches are strongly correlated with each other, as expected because they all scale (almost linearly) with F_ν(1mm). They also correlate well with the CO-based H₂ mass estimates over one and a half dex of dynamic range. However, systematic offsets are observed. The Groves et al. (2015) estimates are on average $1.5\times$ lower than those based on CO. The estimates based on Scoville et al. (2016) are another $2\times \,$ lower, and the masses based on MAGPHYS are a factor $1.7\times \,$ lower than those obtained following Scoville et al. (2016).

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What causes the discrepancies between these mass estimates? The CO masses might be overestimated because of our assumptions in terms of CO excitation and ${\alpha }_{\mathrm{CO}}$. A higher CO excitation would imply higher r_J1, thus lower CO(1–0) luminosity (see Equation (1)). When we assume the M82 excitation template by Weiß et al. (2007) (see Figure 5), the inferred ${M}_{{\rm{H}}2}$ masses would be $1.3\times$ lower for ID.2–6, and $2.2\times$ lower for ID.1 and ID.10. This would solve the discrepancy with respect to the estimates based on the Groves recipe, and it would mitigate, but not solve, the discrepancy with the other gas mass estimates. However, such a high CO excitation scenario is ruled out by our 1 mm line observations (see Figure 5). A lower value of ${\alpha }_{\mathrm{CO}}$ could also help. When we adopt the classical value for ULIRGs, ${\alpha }_{\mathrm{CO}}\,=0.8$ ${M}_{\odot }$(K km s⁻¹ pc²)⁻¹ (Bolatto et al. 2013), the CO-based gas masses would be a factor 4.5 lower, thus in good agreement with those from the dust. Figure 6 shows that the majority of our sources lie along the relation of main-sequence galaxies/local spiral galaxies in the ${L}_{\mathrm{IR}}$–${L}_{\mathrm{CO}}^{\prime }$ plot. This is regardless of the choice of ${\alpha }_{\mathrm{CO}}$. Thorough studies of galaxies along this sequence support our choice for a higher value of ${\alpha }_{\mathrm{CO}}$ (e.g., Daddi et al. 2010a; Genzel et al. 2010, 2015; Sargent et al. 2014). Further support for our choice comes from the position of our sources along the "main sequence" of galaxies (Figure 1). Of the sources listed in Table 4 and appearing in Figure 10, only ID.5 could be considered a starburst in this respect. Adopting a lower ${\alpha }_{\mathrm{CO}}$ for only this source would lower its molecular gas mass by a factor ∼4.5, thus bringing it close to the bulk of the "main-sequence" galaxies in terms of gas fraction (Figure 9), but pushing it away from the sequence in the star formation law plot (Figure 7). It would also reduce its depletion timescale (Figure 8) and bring the CO-based gas mass closer to the dust-based estimates (Figure 10). Similar considerations could also apply for ID.1, the CO-brightest galaxy in our sample. The compact morphology and the small separation from a companion galaxy, the rising CO emission at high J, the high values of ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ and ${{\rm{\Sigma }}}_{{\rm{H}}2}$, and the very high ${M}_{{\rm{H}}2}$/M_* all point toward a starburst scenario for this source; however, it is located along the main sequence of galaxies at $z\sim 2.5$ in Figure 1, and the ${L}_{\mathrm{IR}}$–${L}_{\mathrm{CO}(1-0)}^{\prime }$ plot (Figure 6) shows that this source is located along the sequence of local spirals and main-sequence galaxies (not along the sequence of starbursts), regardless of the choice of ${\alpha }_{\mathrm{CO}}$, even when we assume the extreme case of thermalized CO(3–2) emission in order to derive ${L}_{\mathrm{CO}(1-0)}^{\prime }$. Because of this, and because of the lack of any starburst signature (justifying a low ${\alpha }_{\mathrm{CO}}$) in all the other sources, the discrepancies between different gas mass estimates shown in Figure 10 cannot be mended solely by tuning our assumptions on the CO-based mass estimates.

The dust-based gas-mass estimates could also be affected by systematic uncertainties. The offset between the estimates based on Equations (2) and (3) suggests a systematic in the calibration of the two recipes. The luminosity range used in Groves et al. (2015) to derive Equation (2), e.g., does not cover the $\gt {10}^{10}$ ${L}_{\odot }$ range, where our galaxies are found. Equation (3), based on Scoville et al. (2014), is pinned down to a longer wavelength than what is observed in ASPECS (850 μm in the rest-frame, instead of ∼500 μm), thus the k correction is significant and dependent on the adopted dust template. In particular, Equation (3) explicitly assumes β = 1.8, which might not be universally valid (see discussion in Paper II). Our dust SED is only poorly sampled. Most remarkably, the comparison between maps of the CO and dust emission in ID.2 suggests that the gas is optically thick over a large area, while the dust is not. We might be missing part of the dust emission as a result of the surface brightness limits, thus affecting our estimates of the total ISM mass. In ID.3, the dust continuum emission is not detected at all, despite the bright CO emission. Since we do not detect any significant 1 mm continuum associated with the extended disk of ID.2, and no dust emission at all in ID.3, it is hard to assess how large a correction we should consider. It is possible that a similar issue is present in other sources, in particular in galaxies that we see as CO emitters, but for which we do not recover any 1 mm continuum emission (see, e.g., Figure 6). Finally, the underlying assumption in the dust-based gas estimates is the dust-to-gas ratio. This can change significantly as a function of metallicity and of other parameters in the galaxy (see Sandstrom et al. 2013, for a detailed discussion). A lower value of DGR (e.g., DGR ∼ 1/200) would halve the discrepancy between the MAGPHYS-based estimates and the CO-based ones. While this is a possibility at the low-mass end of Figure 10, we point out that the relatively high stellar mass of the galaxies at the bright end supports metallicity values close to solar, thus disfavoring the high DGR values needed to reconcile the two gas-mass estimates.