THE KILOPARSEC-SCALE STAR FORMATION LAW AT REDSHIFT 4: WIDESPREAD, HIGHLY EFFICIENT STAR FORMATION IN THE DUST-OBSCURED STARBURST GALAXY GN20*

J. A. Hodge; D. Riechers; R. Decarli; F. Walter; C. L. Carilli; E. Daddi; H. Dannerbauer

doi:10.1088/2041-8205/798/1/L18

1. INTRODUCTION

One of the most commonly used diagnostics in studies of galaxy formation is the relation between gas surface density (Σ_gas) and star formation rate (SFR) surface density (Σ_SFR), which describes the relative efficiency at which gas is transformed into stars in different environments (e.g., Schmidt 1959; Kennicutt 1998a). Recent studies indicate that this star formation (SF) law (or "Kennicutt–Schmidt" law) is molecular (e.g., Bigiel et al. 2008; Leroy et al. 2008; Schruba et al. 2011), and that it does not evolve with redshift (e.g., Daddi et al. 2010; Genzel et al. 2010). There is evidence, however, for two different SF regimes on the Σ_SFR– ${\rm \Sigma }_{\rm H_2}$ plane ("main-sequence galaxies" versus "starbursts"; Daddi et al. 2010; Genzel et al. 2010, but see Narayanan et al. 2012).

Due to the difficulty of resolving the cold molecular gas and obscured star-forming regions in distant star-forming galaxies, studies of the high-redshift SF law have historically been limited to unresolved detections. However, it is clear that resolved observations are necessary to test whether the gas and SFR surface densities on sub-galactic scales are significantly different than those implied by global averages, as well as whether the most intensely star-forming galaxies at high redshift—which appear to have Σ_SFR that are consistent with theories of so-called "maximum starbursts" (e.g., Elmegreen 1999; Tacconi et al. 2006)—are approaching the Eddington limit for dust in radiation pressure-supported disks (Scoville 2003; Thompson et al. 2005) on ∼kiloparsec or even galaxy-wide scales (e.g., Walter et al. 2009; Riechers et al. 2013). While such resolved studies have become possible in the last few years, they are still limited to either z < 2 (Freundlich et al. 2013) or gravitationally lensed galaxies (Swinbank et al. 2010; Decarli et al. 2012; Sharon et al. 2013; Rawle et al. 2014).

To this end, we here report high-resolution rest-frame far-infrared (FIR) imaging of GN20 (Pope et al. 2006; Daddi et al. 2009), a z = 4.05 submillimeter galaxy (SMG; Casey et al. 2014). GN20 has the deepest, highest-resolution cold molecular gas imaging currently available for any unlensed high-z galaxy, revealing an extended (∼14 ± 4 kpc) gas reservoir resolved on scales of ∼1.3 kpc (Hodge et al. 2012, hereafter H12). The observations presented here provide nearly matched-resolution imaging of the dust emission, allowing us to investigate the SF efficiency (SFE) and SF law on ∼kiloparsec scales 12 Gyr ago. Where applicable we assume a concordance, flat ΛCDM cosmology (H₀ = 71 km s⁻¹ Mpc⁻¹, Ω_Λ = 0.73, Ω_M = 0.27; Spergel et al. 2007).

2. OBSERVATIONS AND DATA REDUCTION

2.1. PdBI 880 μm

The 880 μm (170 μm rest-frame) observations of GN20 (α(J2000) = 12^h37^m11920, δ(J2000) = 62°22'12 farcs 0) were carried out in two tracks on 2013 December 4 (C-configuration track) and 2014 March 7 (A-configuration track) using six antennas. The observations used the Plateau de Bure Interferometer's (PdBI's) Band 4, tuned to 340 GHz (880 μm), along with the WideX correlator (3.6 GHz bandwidth). The receiver was operating in the upper side band. The nearby radio quasars B1044+719 and B1418+546 were used for pointing, amplitude, and phase calibration, and the flux calibrators were 3C279 and MWC349 (A-track) and 3C84 and LKHα101 (C-track). We estimate the flux calibration to be good to within 20%.

The IRAM gildas package was used for data reduction and analysis. The calibration process included two iterations of phase-only self-calibration, and one iteration of amplitude+phase self-calibration. After flagging, there were 5.7 h/4.3 h on-source in the C/A-tracks (six-antenna equivalent). The final map (C+A tracks; Figure 1) has 0 farcs 05 pixels and was created using the clean algorithm with robust weighting and a tight clean box on the source (cleaning down to 2σ). The image has a synthesized beam of 0 farcs 3 × 0 farcs 2 and an rms of 0.25 mJy beam⁻¹. As the source lies at the phase center of the 14 farcs 8 primary beam, the map was not corrected for primary beam attenuation.

**Figure 1.** Multiwavelength imaging of the z = 4.05 SMG GN20. Top left: 880 μm PdBI image (03 × 02 resolution). Contours start at ±2σ in steps of 1σ = 0.25 mJy beam⁻¹ (corresponding to a rest-frame brightness temperature of *T_B* = 0.22 K). The measured peak flux density of 3.6 mJy implies T_{B, peak} = 3.2 K. Top right: VLA CO(2–1) 0th moment map (H12) at the same resolution as the 880 μm data. Bottom left: 1.2 mm PdBI image (046 × 035 resolution). Contours start at ±2σ in steps of 1σ = 0.25 mJy beam⁻¹. Bottom right: *HST*/WFC3 F105W image from the CANDELS survey (Grogin et al. 2011) and 1.2 mm contours.
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2.2. PdBI 1.2 mm

The 1.2 mm (240 μm rest-frame) PdBI observations of GN20 (20 farcs 1 primary beam) were carried out in two tracks on 2010 February 1 and 7 using six antennas in the A-configuration. The 250 GHz observations used the previous generation correlator, providing a total bandwidth of 1.0 GHz (dual polarization). The quasars B1044+719, B1418+546, and B1300+580 were used for pointing, amplitude and phase calibration. Several standard calibrators (MWC349, Titan, 3C273, 3C279, 3C345, 3C454.3, B0234+285, B0923+392, B1055+018, B1749+096) were observed for flux and bandpass calibration, yielding ∼15% calibration accuracy.

The data were mapped using clean with natural weighting, resulting in a synthesized beam of 0 farcs 46 × 0 farcs 35 (Figure 1). Weighting schemes that would lead to higher spatial resolution were discarded due to high noise. The final rms noise is 0.25 mJy beam⁻¹ over the full bandpass.

2.3. VLA CO(2–1)

The CO(2–1) data on GN20 were obtained with the Karl G. Jansky Very Large Array (VLA) and presented in Carilli et al. (2011) and H12. The 0th moment map used here (Figure 1) was created by taking the native resolution (0 farcs 19/1.3 kpc) data cube and convolving it to the resolution of the 880 μm data (0 farcs 3 × 0 farcs 2), applying the same mask as used in H12.

3. RESULTS

3.1. Spatially Resolved Rest-frame FIR Emission

We have detected and spatially resolved the 880 μm (170 μm rest–frame) continuum emission from GN20 (Figure 1). We measure a peak flux density of 3.6 mJy beam⁻¹ at α(J2000)=12^h37^m1190, δ(J2000) = 62°22'12 farcs 10, positionally coincident with the peak of the CO emission within 0 farcs 1, and suggesting that the most intense SF occurs in the region with the highest concentration of cold gas. The integrated flux density (16 ± 1 mJy) is consistent with the integrated flux density measured in the C-configuration data alone (18 mJy; beam 0 farcs 88 × 0 farcs 74), indicating that the higher-resolution data are not resolving out a significant amount of emission. These measurements are also consistent with the single dish measurement for GN20 (S₈₅₀ = 20.3 ± 2.1 mJy; Pope et al. 2006). Note that despite its large flux density, the evidence suggests that GN20 is not lensed (Carilli et al. 2010).

The full size of GN20's molecular gas reservoir is 14 ± 4 kpc (H12). From the new 880 μm continuum data, we find that the dust-obscured SF is also extended, with a bar-like extension along the major axis of the galaxy. We measure a deconvolved source size of (0 farcs 76 ± 0 farcs 06) × (0 farcs 33 ± 0 farcs 03) (FWHM, equivalent to ∼5.3 × 2.3 kpc), implying a physical size of ∼10 kpc out to 10% of the peak flux density (along its major axis). This suggests that the dust-obscured SF in this galaxy is almost as extended as the gas reservoir—a conclusion which is consistent with the spatially resolved excitation structure (H12).

While the FIR emission is coincident with the CO emission, both the FIR and CO emission are offset by ∼0 farcs 6 (4 kpc) from the peak of the rest-frame UV emission as traced by the Hubble Space Telescope (HST)/Wide Field Camera 3 (WFC3) F105W image (Figure 2). The striking anti-correlation suggests that the copious dust surrounding the starburst heavily obscures the UV/optical light produced by newly formed stars and keeps it from escaping the galaxy in all but one small region several kiloparsecs from the nuclear burst.

**Figure 2.** False-color image of GN20, showing the *HST*/WFC3 F105W emission (tracing young stars; blue), the VLA CO(2–1) emission (tracing cold molecular gas; green), and the PdBI 880 μm emission (tracing dust-obscured SF; red).
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We have also detected and resolved GN20 in the 1.2 mm (240 μm rest-frame) continuum observations (Figure 1). We measure an integrated flux density of 5.7 ± 1.6 mJy, which is ∼2/3 of the flux measured with the MAMBO-2 bolometer at 11'' resolution at the same wavelength (9.3 ± 0.9 mJy; Greve et al. 2008). As the 1.2 mm data were taken entirely in the extended configuration, this could indicate that some spatially extended flux on ≳1'' scales is resolved out. The FWHM source size of (0 farcs 77 ± 0 farcs 17) × (0 farcs 49 ± 0 farcs 17) measured from the 1.2 mm observations is consistent with that measured from the 880 μm data.

3.2. Dust Continuum Slope

We used our dust imaging to examine possible deviations from a constant 880 μm/1.2 mm ratio, which could suggest variations in the dust continuum slope across the galaxy. A gradient in the slope could indicate a gradient in the dust temperature and/or optical depth—both of which may be expected for dense nuclear starbursts. We convolved the 880 μm image to the resolution of the 1.2 mm image, resampled both maps to have 0 farcs 4 pixels (∼2.8 kpc, the approximate size of the resolution element), and blanked all pixels except those with flux densities >1σ in both maps in a central contiguous region. The resulting flux densities are plotted in Figure 3, where the solid line represents the ratio expected from the best-fit Draine & Li (2007) dust model to the IR spectral energy distribution (SED)—see Tan et al. (2014, hereafter T14) for further details. In general, we find 1.2 mm flux densities that are ∼20%–25% lower than expected, providing further evidence that the lack of short baseline information may have resolved out some extended emission.⁹ The remaining scatter, however, is not statistically significant. We conclude that the data are consistent with a constant ratio, with no evidence for a variation in the dust continuum slope across the galaxy. This finding is consistent with the apparently extended nature of the SF in GN20, and it supports the use of a constant Σ_SFR/S_880 μm conversion factor in the analysis that follows.

**Figure 3.** S_880 μm vs. S_1.2mm for 04 (∼2.8 kpc) resolution elements. The solid line represents the ratio expected from the best-fit model to GN20's IR SED (Tan et al. 2014).
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3.3. SFR Surface Densities

GN20 has an infrared luminosity of log(L_IR/L_☉) = 13.27 ± 0.02, implying a total SFR of 1860 ± 90 M_☉ yr⁻¹ (assuming a Chabrier initial mass function, IMF; T14). This is consistent with an analysis of the polycyclic aromatic hydrocarbon emission (Riechers et al. 2014), indicating no significant active galactic nucleus contamination to L_IR. Using the extent of the 880 μm emission, the average Σ_SFR is ∼100 M_☉ yr⁻¹ kpc⁻². To determine how Σ_SFR varies on ∼kiloparsec scales, we resampled the 880 μm map onto a grid of 0 farcs 25/1.75 kpc pixels and used the best-fit Draine & Li (2007) dust model to the IR SED (T14) to determine a conversion factor between the 880 μm flux densities and IR luminosities of 1.006(±0.045) × 10¹² L_☉ mJy⁻¹. The uncertainty derives from the uncertainty on the (model-derived) L_IR, which was quantified by T14 using a series of MC simulations. We then divided by the beam area and scaled based on the total SFR/L_IR for GN20. We find values in the ∼tens of M_☉ yr⁻¹ kpc⁻², peaking at 119 ± 8 M_☉ yr⁻¹ kpc⁻² in the galaxy's center.

3.4. The Resolved SF Law at z = 4

In order to study the spatially resolved SF law in GN20, we resampled the CO(2–1) map onto the pixel grid described in Section 3.3. The observed flux densities from the CO and 880 μm maps are plotted in Figure 4 (left). The ratio between FIR and CO emission (indicating SFE, and thus gas depletion timescale) is highest in the central regions of the galaxy, which have the highest gas columns. The SFE tends to decrease further from the galactic center, with an overall variation of ∼8 across the regions where most of the SF takes place. A function of the form Σ_880 μm ∼ (Σ_CO)^N was fit to the data using the IDL routine linmix $\_$ err (Kelly 2007), a linear regression estimator that uses a Bayesian approach and considers measurement errors in two variables. The result of the fit gives a slope of N = 2.1 ± 1.0. This slope assumes a constant Σ_SFR/S_880 μm ratio, as supported by our analysis in Section 3.2. To simulate the effect of a variable dust temperature across the source, we re-ran the Draine & Li models with a wide range of ionization parameters (U_min = 10–100), resulting in Σ_SFR increasing/decreasing by less than a factor of two. Given the radial trend in SFE, this would increase the vertical scatter in the data, steepening the slope.

Figure 4 (right) shows the GN20 data on the Σ_SFR– $\Sigma _{\rm H_2}$ plane. In order to convert the CO(2–1) flux densities to molecular gas masses, we have assumed the standard relationships from Carilli & Walter (2013) and a CO-to-H₂ conversion factor of α_CO = 1.1 M_☉(K km s⁻¹ pc²)⁻¹. This value—which is slightly higher than the typically adopted ULIRG value—was determined in H12 by using the dynamical mass (derived by modeling the CO(2–1) dynamics) to constrain the gas content.¹⁰ The mean depletion time we measure for GN20 is 130 Myr, with a range of 40–300 Myr. A compilation of literature studies is shown for comparison, where all SFR measurements have been converted to a common Chabrier (2003) IMF. The gas masses were calculated using the CO-to-H₂ conversion factors assumed in the respective studies: α_CO = 0.8 for the unresolved ULIRGs/SMGs (including helium; Kennicutt 1998b; Bouché et al. 2007; Bothwell et al. 2010); α_CO = 3.6 for the z ∼ 0.5 disk galaxies and BzKs (Daddi et al. 2010); α_CO = 4.35 for the local spirals and color-selected galaxies (Tacconi et al. 2010; Leroy et al. 2013; Freundlich et al. 2013); and α_CO = 4.6 (Sharon et al. 2013) and α_CO = 0.7 (Rawle et al. 2014) for the two strongly lensed resolved SMGs.¹¹^,¹²

Figure 5 shows Σ_SFR versus $\Sigma _{\rm H_2}$ , where the latter is divided by free-fall time (t_ff). The comparison data are from Krumholz et al. (2012). To calculate t_ff for GN20, we used Equation (8) from Krumholz et al., which is appropriate for high surface density galaxies in the Toomre regime, and we have assumed the Toomre parameter Q = 1, the dimensionless constant ϕ_P = 3 (Krumholz & McKee 2005), the logarithmic index of the rotation curve β = 0 (flat rotation curve; H12), and the angular velocity of galactic rotation Ω = 0.175 Myr⁻¹ (calculated at the half-light radius). For GN20, this gives values of the SFE per free-fall time ( epsilon _ff; i.e., the fraction of gas converted into stars per free-fall time) ranging from epsilon _ff = 0.01–0.18. Also shown is the local volumetric SF law from Krumholz et al. (2012), evaluated at their best-fit value of epsilon _ff = 0.015.

4. DISCUSSION

While it is evident that SMGs host some of the largest starbursts in the known universe (∼10³ M_☉ yr⁻¹), this knowledge is largely based on integrated measurements and/or indirect SFR tracers. The latter imply that SMGs have typical SFR densities of ∼80 M_☉ yr⁻¹ kpc⁻² (Tacconi et al. 2006), well below the ∼1000 M_☉ yr⁻¹ kpc⁻² limit based on a theoretical description of the (dust–opacity) Eddington-limited SF of a radiation pressure-supported starburst on kiloparsec scales (Scoville 2003; Thompson et al. 2005), though exceptions do exist (e.g., ∼600 M_☉ yr⁻¹ kpc⁻² in the z = 6.3 SMG HFLS3; Riechers et al. 2013). Indeed, the highest-resolution existing submillimeter observations on the lensed z ∼ 2.3 "Eyelash" galaxy show that the star-forming regions are undergoing maximal starbursts (Swinbank et al. 2010). Previous 0 farcs 8 890 μm imaging (Younger et al. 2008) of GN20—which is intrinsically much more luminous than the Eyelash—indicated that it might be forming stars close to or at its Eddington limit, although subsequent CO imaging (Carilli et al. 2010) revealed the large gas disk and, along with a revised total SFR (Daddi et al. 2009), implied a sub-Eddington average value. Still, it was unknown whether the SF in GN20 was more compact than the cold gas reservoir traced by CO(2–1), or whether the indirectly derived average value served to mask extreme differences in Σ_SFR between clump/inter-clump regions. Our high-resolution 880 μm imaging shows that the SF in GN20 remains sub-Eddington on scales down to a few kpc².

An examination of the resolved SF law in this unlensed z ∼ 4 galaxy produced a power-law slope of N = 2.1 ± 1.0. While the significant uncertainty on the slope means that it may be only barely steeper than the local linear relation (e.g., Leroy et al. 2013), other studies also find evidence for a steepening in gas-rich/high density environments (i.e., ⩾200 M_☉ pc⁻²; Solomon et al. 1987; Gao & Solomon 2004; Rosolowsky & Blitz 2005), including the centers of nearby galaxies (e.g., Narayanan et al. 2008; Leroy et al. 2013). A slope of N ∼ 1.5 may be expected if SFR is proportional to gas mass divided by free-fall time (assuming a constant scale height; e.g., Madore 1977; Krumholz et al. 2009). Other models of feedback-regulated SF (e.g., Ostriker & Shetty 2011; Faucher-Giguère et al. 2013) give a quadratic slope—closer to that seen here, though it is impossible to differentiate between models given the current observational uncertainties.

When compared with other studies on the Σ_SFR– ${\rm \Sigma }_{\rm H_2}$ plane, we find that the GN20 data lie above the sequence traced out by "normal" star-forming disk galaxies. This is consistent with its classification as a starburst (T14) and, given that the CO(2–1) data show evidence for a smoothly rotating gas disk (H12), it implies that the presence of an ordered velocity field is not a clear indication of the presence or absence of intense starburst activity. The approximate alignment of GN20 and the two (resolved) lensed SMGs is also notable. Given that the conversion factors specifically derived for the individual cases range from ULIRG-like (Rawle et al. 2014) to slightly above ULIRG-like (GN20) to Galactic (Sharon et al. 2013), this suggests that the dispersion in the SF law is apparently not due solely to choice of α_CO. On the contrary, the GN20 results suggest that the dispersion in the SF law is real, and that SMGs can have very high SF efficiencies on small scales.

If one assumes a universal SF law where the SFR volume density is simply a function of the molecular gas volume density and local t_ff (Schmidt 1959; Kennicutt 1998b; Krumholz & McKee 2005; Leroy et al. 2008), then the complexity in the Σ_SFR– $\Sigma _{\rm H_2}$ plane should be removed by accounting for t_ff. Studies substituting the global dynamical timescale (t_dyn) for t_ff in disk galaxies and mergers at low/high redshift (Daddi et al. 2010; Genzel et al. 2010) provided preliminary support for this theory, as t_dyn is proportional to t_ff for galactic scales in high surface density galaxies (Krumholz & McKee 2005; Leroy et al. 2008; Genzel et al. 2010; Krumholz et al. 2012). More recently, Krumholz et al. (2012) presented a complete theoretical framework for a local volumetric SF law, including a model for the local (star-forming cloud) t_ff, which allowed them to describe the SF on scales from entire high-redshift galaxies down to individual molecular clouds with a fixed value of epsilon _ff = 0.015. Figure 5 demonstrates that our resolved imaging of GN20 is also consistent with this model, extending the evidence for a fixed SFE per free-fall time to include the star-forming medium on ∼kiloparsec scales in a galaxy 12 Gyr ago.

We thank John Hibbard, Mark Krumholz, Adam Leroy, Kartik Sheth, and an anonymous referee for helpful comments/discussions, and Roberto Neri, Sabine König, and Melanie Krips for help with data analysis. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). Support for R.D. was provided by the DFG priority program 1573 "The physics of the interstellar medium."

THE KILOPARSEC-SCALE STAR FORMATION LAW AT REDSHIFT 4: WIDESPREAD, HIGHLY EFFICIENT STAR FORMATION IN THE DUST-OBSCURED STARBURST GALAXY GN20^*

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ABSTRACT

1. INTRODUCTION