High Dense Gas Fraction in a Gas-rich Star-forming Galaxy at z = 1.2^∗

We observed the primary target EGS 13004291 (J2000 R.A.: 14^h19^m150, decl.: +52^d49^m30^s, z = 1.197) with the IRAM Plateau de Bure Interferometer (PdBI) with five antennas in the compact D configuration during five tracks in 2010 May–June, with a total on-source time of 11.5 hr. Weather conditions were average or better for 3 mm observations, with a precipitable water vapor of 6–7 mm for all tracks. The absolute flux scale was calibrated either on 3C273, MWC349, or 3C345. The source J1418+546 was used as a phase and bandpass calibrator. The WideX correlator (bandwidth ∼3.6 GHz) was used to observe multiple molecular lines simultaneously. Observations were carried out in dual polarization mode, using a tuning frequency of 81.95 GHz and a binned spectral resolution of 3.9 MHz (∼14.2 km s⁻¹ at 81.95 GHz), covering the frequency range ${\nu }_{\mathrm{obs}}=80.6833\mbox{--}83.4366\,\mathrm{GHz}$. This covered the redshifted HCN(J = 2 → 1), HNC(J = 2 → 1), ${\mathrm{HCO}}^{+}(J=2\to 1)$, HC₃N(J = 20 → 19), and ${{\rm{H}}}_{2}{\rm{O}}({3}_{13}\to {2}_{20}$) lines at z = 1.197 (see Table 1 for rest frequencies).

Table 1.  Observed Line Properties for EGS 13004291

Transition	${\nu }_{\mathrm{rest}}$	${\nu }_{\mathrm{obs}}$	I	$L^{\prime}$	Referencesa
	(GHz)	(GHz)	(Jy km s−1)	(1010 K km s−1 pc2)
CO($J=2\to 1$)	230.5379	104.961	3.09 ± 0.27	6.0 ± 0.5	(1)
CO($J=3\to 2$)	345.7959	157.394	4.6 ± 0.1	3.9	(2)
HCN($J=2\to 1$)	177.2612	80.704	0.28 ± 0.09	0.9 ± 0.3	(1)
HNC($J=2\to 1$)	181.3248	82.555	0.17 ± 0.07	0.5 ± 0.2	(1)
HC3N(J = 20 → 19)b	181.9449	82.837	$\lt 0.21$	$\lt 0.65$	(1)
${\mathrm{HCO}}^{+}(J=2\to 1)$ b	178.3750	81.212	$\lt 0.23$	$\lt 0.76$	(1)
${{\rm{H}}}_{2}{\rm{O}}({3}_{13}\to {2}_{20}$)b	183.3101	83.459	$\lt 0.22$	$\lt 0.66$	(1)

Notes.

^a(1) This work. (2) Tacconi et al. (2013). ^b $3\sigma$ upper limits obtained as described in Section 3.2.

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In a second tuning, we observed the CO($J=2\to 1$) line (${\nu }_{\mathrm{rest}}=230.538$ GHz) in 2010 May for 0.6 hr on source, under excellent weather conditions for 3 mm observing, using six antennas and recording data from both the narrow-band (bandwidth ∼1.0 GHz) and the WideX correlator. We used a tuning frequency of 104.933 GHz and binned spectral resolutions of 10 and 2 MHz, corresponding to 28 km s⁻¹ and 5.6 km s⁻¹ at 105 GHz, respectively. The data were taken in dual polarization mode, using MWC349 as the absolute flux calibrator.

All observations were calibrated using the IRAM PdBI data reduction pipeline in CLIC (Continuum and Line Interferometer Calibration),⁶  with subsequent additional flagging by hand. The absolute flux scale was calibrated to better than 20% for both CO and HCN observations.

The reduced visibility cube for the dense gas tracers was imaged using UV_MAP, with natural weighting and a pixel size of $0\buildrel{\prime\prime}\over{.} 5\times 0\buildrel{\prime\prime}\over{.} 5$, and cleaned using the task CLEAN with the Hogbom algorithm, after binning over a large velocity width, 40 channels at our spectral resolution (corresponding to ∼590 km s⁻¹). The resulting cleaned image has an rms noise of 0.15 mJy beam⁻¹ and a synthesized beam size of $6\buildrel{\prime\prime}\over{.} 0\,\times 5\buildrel{\prime\prime}\over{.} 0$, with a position angle (PA) of −723. The antenna half-power beam width is 61'', and primary beam correction was applied using the task PRIMARY.

The task UV_MAP was used to image the calibrated visibility CO($J=2\to 1$) cube, using natural weighting and a pixel size of $0\buildrel{\prime\prime}\over{.} 5\times 0\buildrel{\prime\prime}\over{.} 5;$ the clean map was made using the task CLEAN, using the Hogbom algorithm in MAPPING with natural weighting. For the CO observations, the cleaned image has an rms noise of 2.3 mJy beam⁻¹ in each channel (width ∼17 km s⁻¹) and a synthesized beam size of $4\buildrel{\prime\prime}\over{.} 7\times 3\buildrel{\prime\prime}\over{.} 8$, with a position angle (PA) of −665. The antenna half-power beam width is 49''.

EGS 13004291 is located in the Extended Groth Strip, and as such has rich multiwavelength coverage from the All-Wavelength Extended Groth Strip International Survey (AEGIS). This includes observations with Chandra (2–10 keV; Laird et al. 2009; Nandra et al. 2015), Galaxy Evolution Explorer (GALEX), Hubble Space Telescope (HST) as part of the 3D-HST survey (Brammer et al. 2012; Skelton et al. 2014), Spitzer IRAC (Barmby et al. 2008), and Herschel PACS/SPIRE observations, as part of the NEWFIRM Medium-band Survey (NMBS; Davis et al. 2007; Whitaker et al. 2011).

To enable a comparison of our source properties against different galaxy populations, we have compiled a large sample of sources from the literature, both local and at high redshift, with extant observations in the FIR, CO (Weiß et al. 2003, 2007; Greve et al. 2005; Riechers et al. 2006b, 2009b, 2011a; Danielson et al. 2011; Thomson et al. 2012), and HCN (Solomon et al. 2003; Vanden Bout et al. 2004; Gao & Solomon 2004a; Isaak et al. 2004; Carilli et al. 2005; Wagg et al. 2005; Evans et al. 2006; Greve et al. 2006; Gao et al. 2007; Graciá-Carpio et al. 2008; Krips et al. 2010; Riechers et al. 2010a; García-Burillo et al. 2012). All obtained luminosities have been adjusted to the cosmology used here; HCN detections with single-dish telescopes pointed at galactic nuclear regions have been treated as lower limits on the HCN luminosity where appropriate. We exclude sources with upper or lower limits on luminosities from all relevant calculations and sample averages, and we only use sources that have solid detections.

We successfully detect the CO($J=2\to 1$) line in EGS 13004291 at a redshift of z = 1.197. We find that the emission is spatially unresolved, consistent with what is expected based on the previously measured CO($J=3\to 2$) size (Tacconi et al. 2013; ${r}_{1/2}\sim 0\buildrel{\prime\prime}\over{.} 5$). We therefore extract the spectral profile (Figure 1) from the peak pixel (J2000 R.A.: 14^h19^m1497; decl.: +52^d49^m2973). The spectrum is fit with a 1D Gaussian to estimate the line peak, velocity width, and central redshift; we find the line center at ${\nu }_{\mathrm{obs}}=104.961\pm 0.005\,\mathrm{GHz}$ corresponding to a redshift of $z=1.1964\pm 0.0001$. The moment-0 map (Figure 1) is created by collapsing the spectral cube along the frequency axis for the FWZI velocity width derived from this best fit (${\rm{\Delta }}{v}_{\mathrm{FWZI}}\sim \,620$ km s⁻¹). Finally, the area under the spectral line fit is used to obtain an integrated line flux of ${I}_{\mathrm{CO}}=3.09\pm 0.27$ Jy km s⁻¹; this is consistent with the value derived from the peak of the moment-0 map. The final spectral profile is shown in Figure 1. From this, we find a CO($J=2\to 1$) luminosity of ${L}_{\mathrm{CO}(J=2\to 1)}^{\prime }=(6.0\pm 0.5)\ \times {10}^{10}$ K km s⁻¹ pc². Assuming a brightness temperature ratio of ${r}_{21}=0.76$ between the $J=2\to 1$ and $J=1\to 0$ transitions (Daddi et al. 2015), we find a CO($J=1\to 0$) luminosity of ${L}_{\mathrm{CO}(J=1\to 0)}^{\prime }=(7.8\pm 0.7)\times {10}^{10}$ K km s⁻¹ pc².

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For each of the lines, a moment-0 map is made using the velocity range −230 to 360 km s⁻¹ (${\rm{\Delta }}v\sim 590$ km s⁻¹), nearly identical to the FWZI velocity range for the CO($J=2\to 1$) line, and the rms noise for each moment-0 map is calculated using the source-free pixels. We tentatively detect the HCN($J=2\to 1$) and HNC($J=2\to 1$) lines (Figure 3), redshifted to ${\nu }_{\mathrm{obs}}=80.704\,\mathrm{GHz}$ and ${\nu }_{\mathrm{obs}}=82.555\,\mathrm{GHz}$, respectively. Given the modest significance of our detections, we use the dirty cube to calculate integrated line fluxes, instead of the cleaned image cube, in order to avoid biases introduced by the cleaning process. We do not make any correction for flux in the side lobes of the dirty beam, given that the source is spatially unresolved. We perform a 2D Gaussian fit to the moment-0 maps of HCN and HNC emission in order to measure the line fluxes. We detect line emission at the source position at significances of $3.0\sigma$ and $2.4\sigma$ for HCN and HNC, with velocity-integrated line fluxes of ${I}_{\mathrm{HCN}}=0.28\pm 0.09$ Jy km s⁻¹ and ${I}_{\mathrm{HNC}}=0.17\pm 0.07$ Jy km s⁻¹, respectively. These correspond to ${L}_{\mathrm{HCN}(J=2\to 1)}^{\prime }$ $\sim \,(9\pm 3)\times {10}^{9}$ K km s⁻¹ pc² and ${L}_{\mathrm{HNC}(J=2\to 1)}^{\prime }$ $\sim \,(5\pm 2)\times {10}^{9}$ K km s⁻¹ pc². The HCN($J\ =2\to 1$) luminosity is extrapolated to the HCN($J=1\to 0$) luminosity using a brightness temperature ratio of ${r}_{21}\sim 0.65$ (Krips et al. 2008; Geach & Papadopoulos 2012) between the $J=2\to 1$ and $J=1\to 0$ transitions (see Section 5.3.2 for a discussion of how this impacts our results), which yields ${L}_{\mathrm{HCN}(J=1\to 0)}^{\prime }$ $\sim \,(1.4\pm 0.5)\times {10}^{10}$ K km s⁻¹ pc².

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We use the same methodology as above to obtain upper limits on ${\mathrm{HCO}}^{+}(J=2\to 1)$, HC₃N(J = 20 → 19), and ${{\rm{H}}}_{2}{\rm{O}}({3}_{13}\to {2}_{20}$) emission. We find $I\lesssim 0.2$ Jy km s⁻¹ pc² for each of the lines, yielding line luminosities $L^{\prime} \ \lt 7$–$8\times {10}^{9}$ K km s⁻¹ pc² (Table 1). These limits are consistent with the limits derived from stacking, as described in Section 4.1. No continuum emission was detected in the CO or dense gas observations, and we set a $3\sigma$ upper limit of $\lt 0.3$ mJy beam⁻¹ on the continuum flux of the source at an observed frame wavelength of 3 mm.

Since both the HCN and HNC lines are only tentatively detected at $\lesssim 3\sigma$ significance, we stack these lines together using both image- and uv-plane stacking to further investigate the reliability of their detection. We stack 200 velocity channels (∼2900 km s⁻¹) around the predicted line peaks (using ${\nu }_{\mathrm{obs}}$ from Table 1 and z = 1.197) for the HCN and HNC($J=2\to 1$) lines. For stacking in the visibility plane, the visibilities $V(u,v)$ are concatenated after the $u,v$ values are scaled by ${\nu }_{i}/{\nu }_{0}$, where ${\nu }_{i}$ is the frequency of channel i, and ${\nu }_{0}$ is the central frequency of observation. Dirty maps from the stacked visibilities are made using UV_MAP. For image-plane stacking, the dirty spectral cubes for individual lines are stacked over the same channel width, centered on the expected HCN and HNC line centers.

The spectral cubes, following both uv-plane and image-plane stacking, are binned over the velocity range ${\rm{\Delta }}v\sim 590$ km s⁻¹ around the central channel, assuming that the velocity widths of CO, HCN, and HNC are the same. Consistency between the two methods is validated by comparing the rms noise achieved after stacking. In addition, the weighted average of HCN and HNC line fluxes, as estimated in Section 3.2, is consistent with the line flux detected in the stacked HCN and HNC emission map. The resulting map (Figure 3) has an rms noise of ∼0.1 mJy beam⁻¹, and an unresolved source is detected at the CO and optical position of EGS 13004291, with a peak flux of 0.4 mJy and at a $4\sigma$ significance.

To ensure the validity of the approach, the same stacking routine is performed using central channels that have been randomly selected, rather than centered on known line positions. Dirty maps are made from the stacked visibilities and searched for significant features. The rms noise is calculated per channel. The number of features found is consistent with the distribution of the peak of a set of generated Gaussian images with an rms noise of 0.1 mJy beam⁻¹ in each channel, as checked using a two-sided Kolmogorov–Smirnov test. In a further 1000 trials, no features at an equal or higher significance are found within a synthesized beam size located at the central pixel.

Finally, we also attempted to stack other dense gas tracers with HCN, and we found that the significance of the detection is reduced in the stacked images. Removing the HCN contribution to the stacked image, we obtain $3\sigma$ upper limits on the HCO⁺, HC₃N, and H₂O line fluxes, consistent with those determined in Section 3.2. We emphasize that for all of our dense gas observations, we use stacking to confirm our results; however, our line fluxes and upper limits are determined for each dense gas tracer individually, as described in Section 3.2.

Given the large bandwidth of the WideX correlator, we perform blind matched filtering on the reduced HCN image cube to search for other spectral lines in the primary beam. We use the code developed by R. Pavesi et al. (2017, in preparation), which performs a blind search for spectral line features in interferometric data cubes. For a given source spatial extent and spectral line width, we generate a template spectral cube, assuming a 2D circular Gaussian to describe the source structure and a 1D Gaussian to describe the spectral line. Due to the spatial correlation of the noise, inherent in interferometric images, the source size is not trivially identical to the "optimal" template size, but requires smaller templates instead (see Pavesi et al. for details). We then convolve the templates defined in this way with the observed data cube, and the resulting cube is then searched for significant features.

We perform the search for a range of possible template spatial sizes ($5^{\prime\prime} \lt {\rm{\Delta }}x\lt 17^{\prime\prime}$) and velocity widths (100 km s⁻¹ $\lesssim \,{\rm{\Delta }}v\lesssim 500$ km s⁻¹). While the same spectral feature might be detected at high significance in multiple templates with varying spatial extents, we expect the highest signal-to-noise ratio (S/N) when the source extent (after convolving with the telescope beam) "matches" the template size, and we therefore select the most significant detection of the source among all the templates. Finally, we obtain the spatial coordinates, peak channel, and significance of the features with the highest S/N. We visually check this list of features, removing obvious contaminants such as bad channels and those features thatoccur at the edge of the image or of the spectral band. Finally, we check the remaining putative features for counterparts in the 3D-HST WFC3-selected photometric catalog of the AEGIS field and the multiwavelength AEGIS catalog (Davis et al. 2007; Whitaker et al. 2011; Brammer et al. 2012; Skelton et al. 2014). Here we discuss the two most likely candidates, both of which are detected at $\geqslant 6\sigma$ significance.

4.2.1. EGS J141917.4+524922

We detect a spectral feature at nearly ∼7σ significance at ${\nu }_{\mathrm{obs}}=82.289\,\mathrm{GHz}$. We fit the spectrum with a 1D Gaussian (Figure 4). We find a FWHM velocity width of ${\rm{\Delta }}{v}_{\mathrm{FWHM}}=481\pm 112$ km s⁻¹ (Table 2). We fit a 2D Gaussian to the moment-0 map for our detected spectral feature to obtain the peak emission position (J2000 R.A.: 14^h19^m17406, decl.: 52^d49^m21155), which corresponds to a distance of $23\buildrel{\prime\prime}\over{.} 9$ from the phase center of our observations. Based on the AEGIS source catalog and Spitzer IRAC and optical counterparts in the HST-CANDELS field, we identify EGS J141917.4+524922 as the nearest counterpart (hereafter J141917+524922). This source is classified as a starburst-dominated ULIRG in the literature, with a known redshift of $z=1.80\pm 0.02$ based on previously obtained Spitzer Infrared Spectrograph (IRS) spectroscopy (Huang et al. 2009). The detected emission thus is consistent with CO($J=2\to 1$) emission at $z=1.8016\pm 0.0002$. Comparing our emission peak position to the optical H-band position for J141917+524922, we find a spatial offset of ∼$0\buildrel{\prime\prime}\over{.} 46\pm 0\buildrel{\prime\prime}\over{.} 41$, which is much smaller than our beam size ($6\buildrel{\prime\prime}\over{.} 0\times 5\buildrel{\prime\prime}\over{.} 0$). We obtain a CO velocity-integrated line flux of ${I}_{\mathrm{CO}}=0.75\,\pm 0.11$ Jy km s⁻¹, after correcting for the primary beam response (Table 2).

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Table 2.  Observation Summary for CO Lines

Source	z	Transition	${\rm{\Delta }}{v}_{\mathrm{FWHM}}$	${I}_{\mathrm{CO}}$	${L}_{\mathrm{CO}}^{\prime }$	References
			(km s−1)	(Jy km s−1)	(1010 K km s−1 pc2)
EGS 13004291	1.1964 ± 0.0001	$J=2\to 1$	340 ± 32	3.09 ± 0.27	6.0 ± 0.5	(1)
	1.197	$J=3\to 2$	311	4.6 ± 0.1	3.9 ± 0.1	(2)
J141917+524921	1.8016 ± 0.0002	$J=2\to 1$	481 ± 112	0.75 ± 0.11	3.1 ± 0.5	(1)
J141912+524924	3.2206 ± 0.0002	$J=3\to 2$	233 ± 54	0.59 ± 0.09	3.0 ± 0.5	(1)

Note. Details of observed CO transitions. (1) This work. (2) Tacconi et al. (2013).

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4.2.2. EGSIRAC J141912.03+524924.0

We detect a spectral feature at ∼6.5σ significance at ${\nu }_{\mathrm{obs}}=81.930\,\mathrm{GHz}$. We fit the spectrum with a 1D Gaussian (Figure 5). We find an FWHM velocity width of ${\rm{\Delta }}{v}_{\mathrm{FWHM}}\sim 233\pm 54$ km s⁻¹ (Table 2). We fit a 2D Gaussian to the moment-0 map for our detected spectral feature to obtain the peak emission position (J2000 R.A.: 14^h19^m12088, decl.: 52^d49^m24235), which corresponds to a distance of $27\buildrel{\prime\prime}\over{.} 0$ from the phase center of our observations. Based on the AEGIS and the 3D-HST/CANDELS catalogs for the AEGIS field, we identify EGSIRAC J141912.03+524924.0 as the nearest counterpart (hereafter J141912+524924). Comparing the emission peak position to the optical H-band position for J141912+524924, we find an offset of ∼$0\buildrel{\prime\prime}\over{.} 57\pm 0\buildrel{\prime\prime}\over{.} 55$, which is much smaller than our beam size ($6\buildrel{\prime\prime}\over{.} 0\times 5\buildrel{\prime\prime}\over{.} 0$).

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Unfortunately, there is no known redshift for this source to allow for immediate spectroscopic confirmation, although it has photometric coverage as part of the AEGIS catalog (Whitaker et al. 2011). We therefore use the photometric redshift code EAZY (Brammer et al. 2008) to estimate a photometric redshift of ${z}_{\mathrm{phot}}\sim 3.23$. We test the robustness of this estimate using a variety of different population synthesis models and find $2.6\lt {z}_{\mathrm{phot}}\lt 4.2$. The emission feature thus is most consistent with the CO($J=3\to 2$) line at $z=3.2206\pm 0.0002$. We obtain a velocity-integrated line flux of ${I}_{\mathrm{CO}}=0.59\ \pm 0.09$ Jy km s⁻¹, after correcting for the primary beam response (Table 2).

We perform SED fitting for all our detected sources using both CIGALE (Code Investigating GALaxy Emission; Noll et al. 2009; Serra et al. 2011) and the high-z extension of MAGPHYS (Multi-wavelength Analysis of Galaxy Physical Properties; da Cunha et al. 2008, 2015). We use all available photometric data points for each of the sources, as described in Section 2.2. Herschel/PACS and Herschel/SPIRE fluxes from the NMBS are available only for EGS 13004291 and J141917+524921, as J141912+524924 is not detected. We used the deblended Herschel fluxes from the NMBS (Whitaker et al. 2011). Since the two serendipitously detected sources are at distances of $23^{\prime\prime}$ and $27^{\prime\prime}$ from EGS 13004291, respectively, and can potentially contaminate its SPIRE fluxes (SPIRE beam size at 350 μm $\sim 24^{\prime\prime}$), the Herschel Interactive Processing Environment (HIPE) software package was used to examine the photometric images. The HIPE tasks SourceExtractorTimeline and source extractors were used to ensure that the deblended FIR fluxes were not contaminated.

CIGALE builds galaxy SEDs from UV to radio wavelengths assuming a combination of modules. These allow us to model the star formation history (SFH), the stellar emission using population synthesis models (Bruzual & Charlot 2003; Maraston 2005), nebular lines, dust attenuation (e.g., Calzetti et al. 2000), dust emission (e.g., Draine & Li 2007; Casey 2012), contribution from an AGN (Fritz et al. 2006; Dale et al. 2014), and radio emission. The SEDs are built while maintaining consistency between UV dust attenuation and FIR emission from the dust. We use simple analytical functions to model the star formation histories—a double exponentially decreasing SFH, and a delayed SFH. We use the dust attenuation from Calzetti et al. (2000) and the dust emission models from Dale et al. (2014). Finally, CIGALE performs a probability distribution function analysis for our specified model parameters and obtains the likelihood-weighted mean value for each.

MAGPHYS similarly uses a Bayesian approach to constrain galaxy-wide physical properties, including the star formation rate, stellar and dust mass, and dust temperature. It builds a large library of reference spectra with different star formation histories (using stellar population synthesis models from Bruzual & Charlot 2003) and dust attenuation properties (using models from Charlot & Fall 2000). It also ensures energy balance between the optical and UV extinction and the FIR emission due to dust.

In order to account for possible AGN contamination in our FIR-luminous sources, we carry out SED fitting with both CIGALE and MAGPHYS for EGS 13004291 and J141917+524922. EGS 13004291 has been detected in the $0.5\mbox{--}2\,\mathrm{keV}$ soft X-ray band in deep Chandra observations of the AEGIS field (Laird et al. 2009; Nandra et al. 2015), raising the possibility that an AGN may be present and that it may contribute to the galaxy's FIR luminosity. However, we can rule out the presence of an AGN for two reasons. First, the galaxy is consistent with the FIR–X-ray correlation (Symeonidis et al. 2011). Second, the hardness ratio of the X-ray emission indicates a starburst origin of the X-ray emission rather than an AGN (Symeonidis et al. 2014). We further test for the presence of an AGN in both our sources by including the fractional contribution of an AGN to the IR luminosity as a free parameter in SED fitting with CIGALE; no good fit is found with a nonzero AGN contribution.

SED fits found using MAGPHYS and CIGALE were consistent within the errors for EGS 13004291 and J141912+524924. We did not find a good fit for the dust peak for J141917+524922 using MAGPHYS. The best-fit SEDs from CIGALE are shown in Figure 6, and the corresponding fit parameters are listed in Table 3. The ${L}_{\mathrm{FIR}}$ for all sources was obtained by integrating the area under the best-fit SED over ${\lambda }_{\mathrm{rest}}=42.5\mbox{--}122.5$ μm, and ${L}_{\mathrm{IR}}$ was obtained by integrating over ${\lambda }_{\mathrm{rest}}=8\mbox{--}1000$ μm (Casey et al. 2014).

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Table 3.  SED-fitting Results and Other Physical Parameters

Source	z	${L}_{\mathrm{FIR}}$	${M}_{\mathrm{dust}}$	${M}_{\mathrm{gas}}$	M*	SFR	sSFR	${\delta }_{\mathrm{GDR}}$	${f}_{\mathrm{gas}}$
		(${10}^{11}{L}_{\odot }$)	(109 M⊙)	(1011 M⊙)	(1011 M⊙)	(M⊙ yr−1)	(Gyr−1)
EGS 13004291	1.197	12.57 ± 4.2	2.4 ± 0.2	2.8 ± 0.3	1.0 ± 0.1	714 ± 42	6.8 ± 0.7	117 ± 21	0.74
J141917+524922	1.802	3.15 ± 1.1	3.7 ± 3.6	1.5 ± 0.2	2.5 ± 0.2	384 ± 40	1.6 ± 0.5	40 ± 44	0.37
J141912+524924	3.221	1.69 ± 0.5	0.7 ± 0.5	2.6 ± 0.4	0.5 ± 0.1	110 ± 7	2.2 ± 0.3	380 ± 342	0.83

Notes. ${L}_{\mathrm{FIR}}$ is calculated by integrating the best-fit SED model over ${\lambda }_{\mathrm{rest}}=42.5\mbox{--}122.5$ μm. The gas masses were calculated using ${\alpha }_{\mathrm{CO}}=3.6{M}_{\odot }$ (K km s⁻¹ ${\mathrm{pc}}^{2}{)}^{-1}$ (see Section 5.2) and the luminosities listed in Table 1, assuming brightness temperature ratios of ${r}_{21}=0.76$ and ${r}_{32}=0.42$.

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Table 4.  Comparison of Source Properties

Source	${L}_{\mathrm{FIR}}$/${L}_{\mathrm{HCN}}^{\prime }$	${L}_{\mathrm{FIR}}$/${L}_{\mathrm{CO}}^{\prime }$	${L}_{\mathrm{HCN}}^{\prime }$/${L}_{\mathrm{CO}}^{\prime }$
	(${L}_{\odot }/{L}_{l})$ a	(${L}_{\odot }/{L}_{l})$ a
EGS 13004291	90 ± 30	16 ± 5	0.17 ± 0.07
Other high-z galaxies:
F10214+4724	2600 ± 550	523 ± 130	0.18 ± 0.04
Cloverleaf	1300 ± 300	135 ± 33	0.10 ± 0.02
Averages:
high-z (U)LIRGs (5)	1667 ± 688	251 ± 129	0.16 ± 0.07
$z\approx 0$ ULIRGs (5)	1347 ± 264	148 ± 33	0.13 ± 0.03
$z\approx 0$ LIRGs (85)	829 ± 791	40 ± 70	0.05 ± 0.04
$z\approx 0$ spirals (45)	618 ± 395	23 ± 23	0.04 ± 0.02

Note.

^a ${L}_{l}={\rm{K}}$ km s⁻¹ pc²; average values were calculated from the archival data as described in Section 2.2.

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We compare our SED-fitting-derived physical parameters (Table 3) against those available in the literature. For EGS 13004291, Tacconi et al. (2013) have derived a star formation rate of SFR ∼ $(630\pm 220)$ M_⊙ yr⁻¹ and a stellar mass of ${M}_{* }\sim (9.3\pm 2.8)\times {10}^{10}{M}_{\odot }$. These values were obtained using an extinction-corrected 24 μm derived IR luminosity + UV flux. However, Freundlich et al. (2013) derive an SFR ∼ 182 M_⊙ yr⁻¹, using the optical [O ii] line luminosity, and a stellar mass of ${M}_{* }\sim 5.0\times {10}^{11}{M}_{\odot }$. They add the caveat that this method is likely to underestimate the SFR, since it does not account for dust-embedded star-forming regions.

Our derived properties from CIGALE depend significantly on the assumed SFH for EGS 13004291. Assuming a stellar history including a young stellar population from recent starburst activity, we find an SFR ∼ $714\pm 42$ M_⊙ yr⁻¹, averaged over the last 10 Myr, and a stellar mass of ${M}_{* }\sim 1.0\times {10}^{11}{M}_{\odot }$. Using only the ${L}_{\mathrm{FIR}}$ as a star formation indicator (Kennicutt 1998), we obtain an SFR ∼ 220 M_⊙ yr⁻¹, which is comparable to the SFR obtained using UV emission; the sum of the SFRs obtained from UV and FIR emission is then roughly consistent with the SFR from SED fitting.

For EGS 141917+524922, previous SED fitting has been performed to the IR and radio photometry (Huang et al. 2009), on which basis it was classified as a starburst-dominated ULIRG. Our stellar mass of ${M}_{* }\sim (2.5\pm 0.2)\times {10}^{11}$ is consistent with the literature value of ${M}_{* }\sim 3\times {10}^{11}$. However, we find ${L}_{\mathrm{IR}}$ ∼ $(9\pm 3)\times {10}^{11}{L}_{\odot }$, which is $5\times$ lower than the literature value. We expect the uncertainties on our estimate to be smaller due to better constraints on the IR luminosity provided by the Herschel/SPIRE photometry, which samples the peak of the FIR emission. Using only the ${L}_{\mathrm{FIR}}$, we obtain an SFR ∼ 96 M_⊙ yr⁻¹, while we obtain an SFR ∼ $(384\pm 40){M}_{\odot }$ yr⁻¹ from SED fitting; as the SED fitting incorporates both the UV and FIR photometry, this implies that the dust-obscured star formation plays a relatively minor role.

Similarly for EGS 141912+524924, we find an SFR ∼ 30 M_⊙ yr⁻¹ using only the ${L}_{\mathrm{FIR}}$; this is significantly smaller than the SFR estimated using complete optical photometric data SFR $(\sim 110\pm 7)$ M_⊙ yr⁻¹, indicating that most of the star formation is not dust-obscured.

The quiescent mode of gas consumption in SFGs at high redshifts can result in markedly different molecular gas excitation than in the ULIRG/SMG/QSO population at comparable redshifts. While some SMGs show a low-excitation molecular gas component (e.g., Ivison et al. 2011; Riechers et al. 2011b), and although there is significant scatter in their excitation properties (e.g., Sharon et al. 2016), the CO excitation of the brightest ULIRGs and SMGs can be nearly thermalized up to J = 3, and mid-J CO lines can be used to trace the bulk of the molecular gas, which is in a warm dense medium in many of these galaxies (e.g., Riechers et al. 2006b, 2009a, 2013). However, the bulk of the molecular gas in MS galaxies lies in extended, cold, diffuse gas reservoirs and can be most reliably traced using low-excitation molecular gas lines, as shown by the subthermal CO excitation prevalent in BzK galaxies (massive, optically selected SFGs at high redshift; see Dannerbauer et al. 2009; Daddi et al. 2010a, 2015). For a limited sample of BzK galaxies, there is also evidence for the presence of an additional warm and dense molecular gas component, resulting in significant CO($J=5\to 4$) emission, which has been suggested to result from giant, dense starbursting clumps (Freundlich et al. 2013; Bournaud et al. 2015; Daddi et al. 2015). However, the observed warm, dense molecular gas component traced by high-J CO lines does not encompass the cold, dense molecular gas traced by low-J HCN emission, which is critical for star formation.

We measure the velocity-integrated line intensity for CO($J=2\to 1$) to be ${I}_{\mathrm{CO}(J=2\to 1)}=3.09\pm 0.27$ Jy km s⁻¹, resulting in a line luminosity of ${L}_{\mathrm{CO}(J=2\to 1)}^{\prime }=(6.0\ \pm 0.5)\times {10}^{10}$ K km s⁻¹ pc² (Figure 1). Based on the ${L}_{\mathrm{CO}(J=3\to 2)}^{\prime }$ value in the literature (Tacconi et al. 2013), we obtain a brightness temperature ratio of r₃₂ ∼ $0.65\pm 0.15$. This is consistent with the average ${r}_{32}=0.58\pm 0.16$ for high-z BzK galaxies (Aravena et al. 2014; Daddi et al. 2015), all of which show significantly subthermal molecular gas excitation at $J\geqslant 3$. This is, however, in sharp contrast to high-z ULIRGs such as the Cloverleaf quasar and F10214+4724, which display a molecular gas excitation consistent with a single high-temperature medium; that is, the CO rotational lines are thermalized to high-J values (${r}_{32}\sim 1;$ Riechers et al. 2011a).

The conversion factor ${\alpha }_{\mathrm{CO}}$ used to derive the molecular gas mass from the CO line luminosity generally depends on the metallicity, the average gas density, the relative fractions of warm and cold dense gas, and the gas excitation in each source (see Solomon & Vanden Bout 2005; Bolatto et al. 2013; Carilli & Walter 2013, for reviews), and different values are appropriate for ULIRGs as opposed to normal SFGs. Here, we attempt to determine the most appropriate ${\alpha }_{\mathrm{CO}}$ for EGS 13004291, based on its dynamical mass. We obtain an estimate for the dynamical mass based on our observed FWHM velocity width ${\rm{\Delta }}{v}_{\mathrm{FWHM}}=340\pm 40$ km s⁻¹ and the previously observed CO($J=3\to 2$) half-light radius ${r}_{1/2}=3.9\ \pm 1.0\,\mathrm{kpc}$, as determined by Tacconi et al. (2013). In addition, we consider the source morphology and orientation. The position–velocity (PV) diagrams for EGS 13004291, using both optical [O ii] and CO spectral line observations, appear to show a disk-like velocity profile (Freundlich et al. 2013). Assuming a disk-like structure for EGS 13004291, we use a 2D Gaussian fitting to the optical HST H-band image to find an axial ratio of $b/a\sim 0.7$, that is, ${\sin }^{2}(i)\sim 0.5$. We then assume a dynamical mass model for a rotating disk, which can be described by ${M}_{\mathrm{dyn}}$ ${\sin }^{2}(i)=233.5$ ${r}_{1/2}{({\rm{\Delta }}{v}_{\mathrm{FWHM}})}^{2}$, where ${\rm{\Delta }}{v}_{\mathrm{FWHM}}$ is in km s⁻¹ and ${r}_{1/2}$ is the half-light radius of the molecular disk in pc (Solomon & Vanden Bout 2005). We thus obtain a dynamical mass of ${M}_{\mathrm{dyn}}=(21\pm 15)\times {10}^{10}{M}_{\odot }$.

Based on the relation between the dynamical mass within the half-light radius, gas mass, and the assumed dark matter (DM) mass, ${M}_{\mathrm{dyn}}=0.5({M}_{* }+{M}_{\mathrm{gas}})+{M}_{\mathrm{DM}}$ (Daddi et al. 2010b), we then calculate the gas mass to be ${M}_{\mathrm{gas}}=(22\pm 15)\times {10}^{10}{M}_{\odot }$ (assuming that DM contributes 25% to the dynamical mass). This implies a CO luminosity to H₂ gas mass conversion factor of ${\alpha }_{\mathrm{CO}}\sim (2.8\pm 2.0)$ M_⊙(K km s⁻¹ ${\mathrm{pc}}^{2}{)}^{-1}$, consistent with the ${\alpha }_{\mathrm{CO}}=3.6$ M_⊙(K km s⁻¹ ${\mathrm{pc}}^{2}{)}^{-1}$ typically used for high-z MS galaxies (e.g., Daddi et al. 2015).

However, we caution that the use of the dynamical mass to estimate ${\alpha }_{\mathrm{CO}}$ suffers from systematic uncertainties resulting from the dynamical estimator used. For example, if we assume an isotropic virial estimator (e.g., Pettini et al. 2001; Binney & Tremaine 2008; Engel et al. 2010)

$$\begin{eqnarray}&&{M}_{\mathrm{dyn}}(r\lt {r}_{1/2})=190{r}_{1/2}{({\rm{\Delta }}{v}_{\mathrm{FWHM}})}^{2}\end{eqnarray} \tag{ 1 }$$

using a normalization for a rotating disk at an average inclination such that ${\sin }^{2}(i$) $=0.25$ (see Bothwell et al. 2010; Engel et al. 2010), we obtain a dynamical mass of ${M}_{\mathrm{dyn}}=(8.5\pm 6.1)\times {10}^{10}{M}_{\odot }$ within ${r}_{1/2}\sim 3.9\,\mathrm{kpc}$, which implies a gas mass of ${M}_{\mathrm{gas}}=(2.8\pm 2.0)\times {10}^{10}{M}_{\odot }$ and a conversion factor of ${\alpha }_{\mathrm{CO}}=(0.4\pm 0.3)$ M_⊙(K km s⁻¹ ${\mathrm{pc}}^{2}{)}^{-1}$. This is more consistent with the values typically found in SMGs and nearby ULIRGs, ${\alpha }_{\mathrm{CO}}=0.8{M}_{\odot }$(K km s⁻¹ ${\mathrm{pc}}^{2}{)}^{-1}$ (e.g., Downes & Solomon 1998; Hodge et al. 2012; Riechers et al. 2014). Adjusting the above calculation to our inclination such that ${\sin }^{2}(i)=0.5$ would further decrease the ${M}_{\mathrm{dyn}}$ by a factor of $\times 2$, which would be inconsistent with our stellar mass ${M}_{* }\sim 1.0\times {10}^{11}{M}_{\odot }$. This discrepancy in ${M}_{\mathrm{dyn}}$ estimates highlights the systematic uncertainties inherent to dynamical mass computations. Motivated by the constraints on the dust-to-gas mass ratio as discussed in Section 5.2, we adopt ${\alpha }_{\mathrm{CO}}=3.6$ M_⊙(K km s⁻¹ ${\mathrm{pc}}^{2}{)}^{-1}$. This is consistent with the ${\alpha }_{\mathrm{CO}}$ derived for PHIBSS galaxies (including EGS 13004291) by Carleton et al. (2016) under the assumption of a constant gas depletion timescale.

We here use the SFR, M_*, ${M}_{\mathrm{dust}}$ from the SED fitting (see Table 3), and the ${M}_{\mathrm{gas}}$ determined from our CO observations to compare our sources to the galaxy MS at their respective redshifts. We use the sSFR/sSFR${}_{\mathrm{MS}}$, the gas-to-dust mass ratio ${\delta }_{\mathrm{GDR}}$, and the gas fraction ${f}_{\mathrm{gas}}$ as diagnostics, where sSFR${}_{\mathrm{MS}}$ is the predicted sSFR for the galaxy MS at the source redshift (Whitaker et al. 2012), ${\delta }_{\mathrm{GDR}}={M}_{\mathrm{gas}}/{M}_{\mathrm{dust}}$, and ${f}_{\mathrm{gas}}$ is defined as

$$\begin{eqnarray}&&{f}_{\mathrm{gas}}=\displaystyle \frac{{M}_{\mathrm{gas}}}{{M}_{\mathrm{gas}}+{M}_{* }}.\end{eqnarray} \tag{ 2 }$$

For EGS 13004291, we find an sSFR/sSFR${}_{\mathrm{MS}}$ ∼ $13\pm 3$, consistent with its classification as a starburst galaxy (Tacconi et al. 2013; Genzel et al. 2015). To estimate the dust temperature, we approximate the Draine & Li (2007) dust model by a blackbody multiplied by a power-law opacity and find a dust temperature of ${T}_{\mathrm{dust}}\approx 20{U}_{\min }^{0.15}\sim 30$ K, where ${U}_{\min }$ is the best-fit intensity of the radiation field from SED fitting (Aniano et al. 2012). Assuming an ${\alpha }_{\mathrm{CO}}=3.6{M}_{\odot }$(K km s⁻¹ ${\mathrm{pc}}^{2}{)}^{-1}$, we find a gas mass of ${M}_{\mathrm{gas}}=(2.8\pm 0.3)\times {10}^{11}$ ${M}_{\odot }$ and a gas fraction of ${f}_{\mathrm{gas}}\sim 0.74$, consistent with the literature value (${f}_{\mathrm{gas}}=0.79;$ Tacconi et al. 2013). We also find a gas-to-dust mass ratio of ${\delta }_{\mathrm{GDR}}\sim 120$. Note that ${\delta }_{\mathrm{GDR}}$ strongly depends on the metallicity of the system (e.g., Leroy et al. 2011; Sandstrom et al. 2013; Rémy-Ruyer et al. 2014; Groves et al. 2015) and can vary significantly depending on the assumed dust properties, although typical values for high-z galaxies are close to ${\delta }_{\mathrm{GDR}}\sim 100$ (Casey et al. 2014). For the $z\sim 1.5$ MS galaxy BzK21000, a ${\delta }_{\mathrm{GDR}}\sim 104$ was found (Magdis et al. 2011). In low- and high-metallicity MS galaxies at $z\sim 1.4$, Seko et al. (2016) find ${\delta }_{\mathrm{GDR}}\sim 570$ and ${\delta }_{\mathrm{GDR}}\sim 400$, respectively. However, their derived ${\delta }_{\mathrm{GDR}}$ may be lower by a factor of $2-3\times$ depending on the dust model assumed. Thus, the ${\delta }_{\mathrm{GDR}}$ in EGS 13004291 is consistent with that seen in $z\sim 1\mbox{--}2$ gas-rich galaxies. Alternatively, if we were to assume an ${\alpha }_{\mathrm{CO}}=0.8{M}_{\odot }$(K km s⁻¹ ${\mathrm{pc}}^{2}{)}^{-1}$, we would obtain a gas mass of ${M}_{\mathrm{gas}}=(6\pm 1)\times {10}^{10}$ ${M}_{\odot }$, a gas fraction of ${f}_{\mathrm{gas}}\sim 0.38$, and a gas-to-dust mass ratio of ${\delta }_{\mathrm{GDR}}=26\pm 21$, which is more consistent with the observed gas-to-dust mass ratio ${\delta }_{\mathrm{GDR}}\sim 40$ for dusty SMGs at high redshift (e.g., Santini et al. 2010; Bothwell et al. 2013; Swinbank et al. 2014; Zavala et al. 2015). Here, ${\alpha }_{\mathrm{CO}}$ varies inversely with the CO excitation temperature ${T}_{\mathrm{ex}}$ (Bolatto et al. 2013); that is, lower ${\alpha }_{\mathrm{CO}}$ values are more typically seen in sources such as SMGs and quasars, which also show higher CO excitation (Ivison et al. 2011; Carilli & Walter 2013), while EGS 13004291 displays subthermal CO excitation consistent with other MS galaxies at $z\sim 1.5$ (Daddi et al. 2015). Therefore, we finally adopt an ${\alpha }_{\mathrm{CO}}=3.6{M}_{\odot }$(K km s⁻¹ ${\mathrm{pc}}^{2}{)}^{-1}$ for our ${M}_{\mathrm{gas}}$ and ${f}_{\mathrm{gas}}$ calculations (see Table 3).

For J141917+524922, we find an sSFR/sSFR${}_{\mathrm{MS}}$ ∼ 2.5, which is consistent with the galaxy MS at z = 1.8 within the scatter (Genzel et al. 2015). This motivates our choice of ${\alpha }_{\mathrm{CO}}=3.6{M}_{\odot }$(K km s⁻¹ ${\mathrm{pc}}^{2}{)}^{-1}$, instead of the value typically used for ULIRGs, ${\alpha }_{\mathrm{CO}}=0.8{M}_{\odot }$(K km s⁻¹ ${\mathrm{pc}}^{2}{)}^{-1}$. Similarly, using a brightness temperature ratio ${r}_{21}=0.76$ (Daddi et al. 2015), we obtain a gas mass of ${M}_{\mathrm{gas}}=(1.5\pm 0.2)\times {10}^{11}{M}_{\odot }$ and a gas fraction of ${f}_{\mathrm{gas}}=0.37$, which is again consistent with the galaxy MS at z = 1.8. We obtain a gas-to-dust mass ratio of ${\delta }_{\mathrm{GDR}}=40\pm 44$, which is consistent with the average value found for SMGs. The large uncertainties on the FIR fluxes of J141917+524922 result in a poorly constrained dust mass and therefore gas-to-dust ratio.

Applying similar diagnostics to J141912+524924, we find an sSFR/sSFR${}_{\mathrm{MS}}$ ∼ 0.8, making this source consistent with the galaxy MS at z = 3.22. J141912+524924 is then the highest-redshift unlensed MS galaxy detected to date. We find a gas-to-dust ratio ${\delta }_{\mathrm{GDR}}=380\pm 342$ and a gas fraction of ${f}_{\mathrm{gas}}\sim 0.83$, implying that it is a gas-rich galaxy with possibly low dust content. We emphasize that the lack of FIR photometry for this source leads to a poorly constrained dust mass and hence gas-to-dust ratio.

We use the following set of diagnostics to characterize star formation in EGS 13004291: the fraction of dense, actively star-forming gas ${f}_{\mathrm{dense}}={M}_{\mathrm{dense}}/{M}_{\mathrm{gas}}\,\propto$ ${L}_{\mathrm{HCN}}^{\prime }$/${L}_{\mathrm{CO}}^{\prime }$, the global star formation efficiency of the molecular gas SFE${}_{\mathrm{mol}}=\mathrm{SFR}/{M}_{\mathrm{gas}}\,\propto$ ${L}_{\mathrm{FIR}}$/${L}_{\mathrm{CO}}^{\prime }$, and the star formation efficiency of the dense gas SFE${}_{\mathrm{dense}}=\mathrm{SFR}/{M}_{\mathrm{dense}}\,\propto$ ${L}_{\mathrm{FIR}}$/${L}_{\mathrm{HCN}}^{\prime }$. Unless explicitly stated, we calculate these ratios using line luminosities for the $J=1\to 0$ transition, calculated in Sections 3.1, 3.2. We calculate ${L}_{\mathrm{FIR}}$/${L}_{\mathrm{HCN}}^{\prime }$, ${L}_{\mathrm{HCN}}^{\prime }$/${L}_{\mathrm{CO}}^{\prime }$, and ${L}_{\mathrm{FIR}}$/${L}_{\mathrm{CO}}^{\prime }$ values for the sample of archival sources described in Section 2.2, and we compare our obtained values for EGS 13004291 against those for different galaxy populations using these diagnostics (see Table 4). Our sample of archival sources includes the Cloverleaf quasar and IRAS F10214+4724, which are among the best studied and brightest HCN and CO sources at high redshift. This allows us to place the properties of its star-forming environment in context. We do not compare our source to APM 08279+5255 at z = 3.9, the only other solid HCN detection at high redshift we have excluded this source despite detailed observations because of its unusual gas excitation properties (e.g., Weiß et al. 2007; Riechers et al. 2010b).

5.3.1. Star Formation Models

Two main classes of dense gas star formation models have been proposed to explain the observed ${L}_{\mathrm{FIR}}$–${L}_{\mathrm{HCN}}^{\prime }$ relation (see Figure 7, middle): threshold density models and turbulence-regulated star formation models. Threshold density models assume a fixed SFR per unit gas mass above a certain critical density, that is, a constant SFE${}_{\mathrm{dense}}$. Working under this assumption, the slope of the ${L}_{\mathrm{FIR}}$–${L}_{\mathrm{HCN}}^{\prime }$ relation (≡ SFE${}_{\mathrm{dense}}$) is expected to remain constant, regardless of the star-forming environment or the galaxy type (Gao & Solomon 2004b; Wu et al. 2005, 2010).

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On the other hand, turbulence-regulated star formation models predict that the SFR depends on the properties of the molecular gas cloud as a whole—the gas surface density, the pressure, and the turbulent velocity dispersion (e.g., Krumholz & McKee 2005; Krumholz & Thompson 2007). Based on these properties, they predict that different molecular gas tracers would produce different slopes in the ${L}_{\mathrm{FIR}}$–gas luminosity plot. In their parameterization, SFE${}_{\mathrm{mol}}$ is a function of both the local free-fall timescale, ${\tau }_{\mathrm{ff}}$, and of the ISM turbulence represented by the Mach number ${ \mathcal M }$. At a given ${ \mathcal M }$, the SFE varies according to $1/{\tau }_{\mathrm{ff}}$, and the SFR varies according to $\bar{n}/{\tau }_{\mathrm{ff}}$, where $\bar{n}$ is the average gas density. The observed SFE${}_{\mathrm{mol}}$ then depends on how the critical density (${n}_{\mathrm{crit}}$) of the tracer in question compares to $\bar{n}$, as this determines the local free-fall timescale; that is, ${\tau }_{\mathrm{ff}}$ will be smaller and SFE${}_{\mathrm{mol}}$ will be higher in regions of large density contrasts. For CO observations, which trace all molecular gas, the average gas density is comparable to the critical density and ${\tau }_{\mathrm{ff}}\propto \,{\bar{n}}^{-0.5}$, and the SFR $\propto {\bar{n}}^{1.5}$, which is close to the slope seen in the ${L}_{\mathrm{FIR}}$–${L}_{\mathrm{CO}}^{\prime }$ relation. If the average gas density is much smaller than the tracer's critical density, ${\tau }_{\mathrm{ff}}$ is determined by ${n}_{\mathrm{crit}}$, and the SFE will be nearly constant; that is, the slope of the ${L}_{\mathrm{FIR}}$–gas luminosity relation is close to unity, thus explaining the slope of the ${L}_{\mathrm{FIR}}$–${L}_{\mathrm{HCN}}^{\prime }$ relation. Additionally, the SFE varies inversely with the Mach number ${ \mathcal M }$; that is, increased turbulence in the ISM hinders efficient star formation.

In this model, ${f}_{\mathrm{dense}}$ increases with both the average gas density $\bar{n}$ and ISM turbulence, as a higher Mach number ${ \mathcal M }$ broadens the density probability distribution of ${f}_{\mathrm{dense}}$ and hence increases the mass of the dense gas above the critical density. Finally, SFE${}_{\mathrm{mol}}$ $=\,{f}_{\mathrm{dense}}\times {\mathrm{SFE}}_{\mathrm{dense}};$ we can thus constrain SFE${}_{\mathrm{dense}}$ using this model.

Observationally, the turbulence-regulated star formation scenario is favored. Merging ULIRGs at both high redshift and in the local universe display a higher ${L}_{\mathrm{FIR}}$/${L}_{\mathrm{HCN}}^{\prime }$, which is consistent with the prediction that SFE${}_{\mathrm{dense}}$ increases with $\bar{n}$ (e.g., Riechers et al. 2007; Bussmann et al. 2008; Graciá-Carpio et al. 2008; Juneau et al. 2009; García-Burillo et al. 2012). Spatially resolved studies of the star-forming regions in local galaxies show that ${L}_{\mathrm{FIR}}$/${L}_{\mathrm{HCN}}^{\prime }$ is strongly dependent on environment and anticorrelated with ${L}_{\mathrm{HCN}}^{\prime }$/${L}_{\mathrm{CO}}^{\prime }$ (e.g., Usero et al. 2015; Bigiel et al. 2016). Thus, ${L}_{\mathrm{FIR}}$/${L}_{\mathrm{HCN}}^{\prime }$ is lower in galaxy nuclear regions, which typically show the highest ${L}_{\mathrm{HCN}}^{\prime }$/${L}_{\mathrm{CO}}^{\prime }$, and increases in regions of low ${L}_{\mathrm{HCN}}^{\prime }$/${L}_{\mathrm{CO}}^{\prime }$; the overall ${L}_{\mathrm{CO}}^{\prime }$/${L}_{\mathrm{FIR}}$ remains nearly constant. This has been interpreted as the increase in SFE${}_{\mathrm{dense}}$ in regions of large density contrasts, instead of in regions with large dense gas fractions, which naturally leads to a lower SFE${}_{\mathrm{dense}}$ in galaxy nuclei, despite a large ${f}_{\mathrm{dense}}$ (Bigiel et al. 2016). All these observational results are in conflict with the idea that star formation proceeds with a universal constant efficiency above a certain threshold density.

An additional source of uncertainty is the HCN to dense gas mass conversion factor ${\alpha }_{\mathrm{HCN}}$, such that ${M}_{\mathrm{dense}}={\alpha }_{\mathrm{HCN}}{L}_{\mathrm{HCN}(J=1\to 0)}^{\prime }$. Studies in the literature suggest that the same ${\alpha }_{\mathrm{HCN}}$ is not applicable to mergers or extreme galaxies (e.g., García-Burillo et al. 2012), and that the ${\alpha }_{\mathrm{HCN}}$ in ULIRGs needs to be lower by a factor similar to ${\alpha }_{\mathrm{CO}}$, relative to the Milky Way (MW) canonical values. This has been taken into consideration while comparing the threshold density and turbulence-regulated star formation models (Usero et al. 2015). García-Burillo et al. (2012) also find that threshold density models place much more stringent constraints on the allowed ${\alpha }_{\mathrm{HCN}}$ and ${\alpha }_{\mathrm{CO}}$ values, as opposed to turbulence-regulated models.

5.3.2. Dense Molecular Gas Properties of EGS 13004291

5.3.3. Other Dense Gas Tracers

5.3.4. Sources of Uncertainty

A number of studies have focused on the bimodality of star formation modes in high-z galaxies, with the bulk proceeding quiescently and a small population of starbursts having higher star formation efficiencies and correspondingly shorter gas depletion timescales (e.g., Daddi et al. 2010b; Rodighiero et al. 2011). However, recent work on large samples of MS galaxies and on outliers above the MS may suggest that all SFGs have the same gas depletion timescale ${\tau }_{\mathrm{depl}}$ and therefore the same star formation efficiency (e.g., Scoville et al. 2015). These authors argue that this short, constant depletion time for both MS and above-MS galaxies is due to a different mode of SF prevailing at high redshifts, driven by either compressive and rapid gas motions or galaxy–galaxy mergers, with the dispersive gas motion arising from the rapid accretion needed to maintain the high SFR. This would be a more efficient mode for star formation from existing gas supplies at high redshift, applying to both MS and above-MS galaxies.

EGS 13004291 lies above the MS, with an sSFR/sSFR${}_{\mathrm{MS}}$ ∼ 13. Despite being a starburst, we find that its SFE is not enhanced relative to other MS galaxies; it is consistent within the scatter (see Figure 6(a) by Dessauges-Zavadsky et al. 2015). This fits in with the model described above, where both MS and above-MS galaxies have the same gas depletion timescale. The lack of enhancement in the SFE${}_{\mathrm{mol}}$ is also consistent with turbulence-regulated star formation models, since the relative effects of an enhanced ${f}_{\mathrm{dense}}$ and a lower SFE${}_{\mathrm{dense}}$ cancel each other out. Given that our SFE${}_{\mathrm{mol}}$ is consistent with that found in MS galaxies at the same epoch, our higher dense gas fraction also fits in with a star formation mode where the ISM is compressive and more turbulent; star formation would no longer proceed in isolated dense gas clumps as in the Milky Way, but in a more widespread manner throughout the disk, so the idea of dense gas clumps as discrete units of star formation would no longer be applicable. This is similar to the prevalent picture in ULIRGs (Solomon & Vanden Bout 2005), although in a milder form because it is driven by less extreme processes than merging galaxies.