EXCITATION MECHANISMS FOR HCN(1–0) AND HCO⁺(1–0) IN GALAXIES FROM THE GREAT OBSERVATORIES ALL-SKY LIRG SURVEY^*

As noted, the data presented here were obtained as part of a millimeter survey of (U)LIRGs selected from GOALS. The GOALS sample as a whole consists of all the (U)LIRGs from the Revised Bright Galaxy Sample (i.e., 60 μm flux density greater than 5.24 Jy and L_IR > 10¹¹ L_⊙; Sanders et al. 2003). GOALS ¹⁴ is a multi-wavelength survey aimed at understanding the physical conditions and activity in the most luminous galaxies in the local universe. The data set includes spectroscopic and imaging observations in the infrared from Spitzer (Petric et al. 2011; Inami et al. 2013; Stierwalt et al. 2013, J. M. Mazzarella et al. 2015, in preparation), and Herschel (Diaz-Santos et al. 2013, 2014; Lu et al. 2014), GALEX, and Hubble Space Telescope (HST) UV, optical, and near-infrared imaging (Howell et al. 2010; Haan et al. 2011; Kim et al. 2013, A. S. Evans et al. 2015, in preparation), Chandra X-ray observations (Iwasawa et al. 2011), and a suite of ground-based radio and sub-millimeter observations (e.g., Leroy et al. 2011; Barcos-Muñoz et al. 2015, R. Herrero-Illana et al. 2015, in preparation). These observations collectively trace the obscured and unobscured activity and constrain the structural properties and merger stages of these systems.

The objects in this study were selected from two portions of GOALS; a high-luminosity sample (L_IR ≥ 10^11.4 L_⊙) to capture the most extreme star-forming systems, and a sample of LIRGs with L_IR ≤ 10^11.4 L_⊙ that were selected to be isolated or non-interacting on the basis of their stellar morphology (Stierwalt et al. 2013). The combination of these subsets of GOALS ensures that this sample spans the range of L_IR and merger stages within the GOALS sample as a whole.

The IRAM 30 m Telescope was used with the EMIR receiver to observe an 8 GHz instantaneous bandwidth, tuned to simultaneously capture HCN ($J=1\to 0)$, ${\mathrm{HCO}}^{+}(J=1\to 0)$, the $\mathrm{CCH}(N=1\to 0)$ multiplet, and $\mathrm{HNC}(J=1\to 0)$, thus reducing systematic uncertainties when comparing the fluxes of these lines. The rest frequencies ¹⁵ for these transitions are: ν_rest (HCN (1–0)) = 88.631 GHz, ν_rest (${\mathrm{HCO}}^{+}(1-0)$) = 89.189 GHz, and ν_rest (HNC (1–0)) = 90.663 GHz. For CCH, we adopt a rest frequency that is the simple arithmetic mean of the individual multiplet frequencies, ν_rest (CCH) =87.370 GHz. The beam FWHM for these measurements is ∼28'', corresponding to a linear size of 16.8 kpc at the mean redshift of our sample—thus we obtain galaxy-integrated measurements. Observations were done in wobbler switching mode (∼1' throw, 0.8 s switching) to improve the baseline calibration. In Table 1 we list the pointing coordinates, assumed redshift, observing year & month, integration times, system temperatures, and the backend used, for each source.

Some of our observations use the FTS backend, which consists of 24 individual Fourier transform spectrometer units. In some scans, the FTS units exhibited gain variations, resulting in offsets of the baselines for individual units. For those scans, linear fits were made to the baseline of each unit (masking out the expected locations of spectral lines), and the units were corrected to a common baseline. Scans were combined and exported using the GILDAS/CLASS software. ¹⁶ Further analysis, including a linear baseline subtraction, smoothing, and flux measurement, was done using the pyspeckit software ¹⁷ (Ginsburg & Mirocha 2011). The typical velocity resolution of these smoothed data is 60 MHz, corresponding to roughly 215 km s⁻¹. In Figure 1 we show the reduced spectra for all sources, with the expected positions of the CCH, HCN (1–0), ${\mathrm{HCO}}^{+}(1-0)$, and HNC (1–0) lines marked. Lines were identified visually based on cataloged optical redshifts and total fluxes were determined by integrating under the line. IRAS F16164–0746 had spectra that were clearly shifted from the location expected based on the cataloged redshift. We measured a new redshift of 0.0229 for IRAS F16164–0746, compared to 0.0272, the value from NED.

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We determined uncertainties in the fluxes by calculating the rms, σ_ch, per channel of width dv in the line-free regions of the spectrum, and multiplying by the square root of the number of channels, N_ch covered by each line. We considered a line a detection if the measured flux exceeded $3{\sigma }_{\mathrm{ch}}{dv}\sqrt{{N}_{\mathrm{ch}}}$. For non-detections we calculated a 3σ upper limit by assuming a square line profile with a width half that was that of the detected HCN (1–0) or ${\mathrm{HCO}}^{+}(1-0)$ line, or 200 km s⁻¹ if no lines were detected for that source.

For IRAM 30 m observations, temperatures are reported on the ${T}_{A}^{*}$ scale; these values were converted to a main beam temperature, T_mb = ${T}_{A}^{*}/$(B_eff/F_eff) using the main beam efficiency, B_eff = 0.81, and a forward efficiency, F_eff = 0.95, both measured by IRAM staff on 2013 August 26. ¹⁸ We use a Jy K⁻¹ conversion of 3.906(F_eff/A_eff) = 6.185, where A_eff = 0.6, to convert the reported ${T}_{A}^{*}$ values to flux densities. Line luminosities were computed according to Solomon et al. (1992, their Equation (3)), in units of K km s⁻¹ pc².

As noted, the wide EMIR bandwidth enables simultaneous measurements of HCN (1–0), ${\mathrm{HCO}}^{+}(1-0)$, HNC (1–0), and CCH. We had detection rates of 78%, 76%, 35%, and 37%, for HCN (1–0), ${\mathrm{HCO}}^{+}(1-0)$, HNC (1–0), and CCH, respectively. For detected lines the signal-to-noise weighted mean HNC (1–0)/HCN (1–0) and CCH/HCN (1–0) ratios are 0.5 ± 0.3 and 0.8 ± 0.3, respectively. One source, NGC 6285 has detected CCH emission and no detection of HCN (1–0) or ${\mathrm{HCO}}^{+}(1-0)$; the CCH detection is ∼3.8σ and, if confirmed, will be an interesting and rare example of a source with a CCH brighter than HCN or HCO⁺. Such elevated CCH emission may indicate an overabundance of CCH or a lack of dense molecular gas (Martín et al. 2014). The HCN (1–0)/${\mathrm{HCO}}^{+}(1-0)$ ratio is discussed in more detail in Section 3. Integrated fluxes (or 3σ upper limits) are provided for all lines in Table 2, but we limit the analysis here to the HCN (1–0) and ${\mathrm{HCO}}^{+}(1-0)$ lines. The upper and lower limits in the figures use these 3σ limits. CO (1–0) observations were also obtained as part of this program; these will be presented in R. Herrero-Illana et al. (2015, in preparation).

Table 2. IRAM 30 m Fluxes

Source	Σ Tmb dv (CCH)	Σ Tmb dv (HCN (1–0))	Σ Tmb dv (${\mathrm{HCO}}^{+}(1-0)$)	Σ Tmb dv (HNC (1–0))
	(K km s−1)	(K km s−1)	(K km s−1)	(K km s−1)
NGC 0034	<0.35	0.71 ± 0.12	1.14 ± 0.15	<0.44
MCG –02-01-052	<0.33	0.57 ± 0.13	<0.27	<0.26
MCG –02-01-051	<0.23	0.17 ± 0.05	0.30 ± 0.07	<0.16
IC 1623	1.40 ± 0.14	1.51 ± 0.15	4.00 ± 0.15	0.64 ± 0.13
MCG –03-04-014	0.48 ± 0.12	0.94 ± 0.12	0.93 ± 0.09	0.38 ± 0.12
IRAS 01364–1042	<0.46	<0.71	<0.45	<0.54
IC 214	<0.15	0.36 ± 0.10	0.56 ± 0.12	<0.14
NGC 0958	<0.42	<0.71	<0.58	<0.69
ESO 550-I G025	<0.35	0.45 ± 0.11	0.36 ± 0.11	<0.39
UGC 03094	0.63 ± 0.12	<0.42	0.74 ± 0.14	<0.41
NGC 1797	<0.37	0.72 ± 0.17	0.55 ± 0.12	<0.43
VII Zw 031	<1.00	<0.98	1.26 ± 0.40	<0.96
IRAS F05189–2524	<0.24	0.73 ± 0.13	0.55 ± 0.13	0.39 ± 0.11
IRAS F05187–1017	0.49 ± 0.12	0.75 ± 0.11	0.76 ± 0.14	<0.18
IRAS F06076–2139	<0.60	1.05 ± 0.20	<0.59	<0.50
NGC 2341	0.65 ± 0.21	<0.35	<0.43	<0.60
NGC 2342	0.55 ± 0.11	0.68 ± 0.11	0.75 ± 0.13	<0.22
IRAS 07251–0248	<0.41	0.35 ± 0.11	<0.33	<0.40
NGC 2623	<1.27	2.62 ± 0.48	2.40 ± 0.54	<1.23
IRAS 09111–1007W	<0.36	0.60 ± 0.08	0.83 ± 0.16	<0.25
IRAS 09111–1007E	0.66 ± 0.16 a	<0.36	<0.41	<0.35
UGC 05101	0.99 ± 0.20	2.17 ± 0.20	1.25 ± 0.17	0.92 ± 0.20
CGCG 011–076	<0.51	1.09 ± 0.15	1.20 ± 0.17	<0.35
IRAS F12224–0624	<0.69	<0.55	<0.67	<0.66
CGCG 043–099	<0.55	<0.54	<0.53	<0.53
ESO 507-G 070	<0.48	1.22 ± 0.19	1.87 ± 0.21	0.49 ± 0.13
NGC 5104	<0.67	1.23 ± 0.22	0.74 ± 0.20	<0.41
IC 4280	0.55 ± 0.15	0.95 ± 0.21	1.17 ± 0.23	0.57 ± 0.18
NGC 5257	<0.29	0.39 ± 0.10	0.46 ± 0.09	0.37 ± 0.11
NGC 5258	<0.20	0.43 ± 0.10	0.53 ± 0.07	0.30 ± 0.08
UGC 08739	<0.51	1.30 ± 0.17	0.94 ± 0.19	0.88 ± 0.19
NGC 5331	<0.31	0.67 ± 0.10	0.45 ± 0.10	<0.43
CGCG 247–020	<0.38	0.96 ± 0.15	0.57 ± 0.12	0.35 ± 0.09
IRAS F14348–1447	<0.46	<0.52	<0.37	<0.51
CGCG 049–057	1.57 ± 0.36	3.21 ± 0.32	1.48 ± 0.20	1.70 ± 0.28
NGC 5936	<0.48	1.21 ± 0.16	0.81 ± 0.11	<0.46
ARP 220	4.37 ± 0.62	12.15 ± 0.75	6.02 ± 0.74	7.85 ± 0.60
IRAS F16164–0746	0.65 ± 0.12	0.54 ± 0.11	0.93 ± 0.12	0.32 ± 0.09
CGCG 052–037	0.57 ± 0.12	0.67 ± 0.12	0.74 ± 0.10	0.73 ± 0.13
IRAS F16399–0937	0.66 ± 0.14	0.77 ± 0.10	0.68 ± 0.10	0.38 ± 0.11
NGC 6285	0.38 ± 0.10	<0.36	<0.25	<0.35
NGC 6286	0.93 ± 0.20	1.63 ± 0.22	1.67 ± 0.21	<0.40
IRAS F17138–1017	1.01 ± 0.17	0.83 ± 0.10	1.03 ± 0.12	0.47 ± 0.11
UGC 11041	<0.50	1.51 ± 0.26	1.49 ± 0.20	<0.59
CGCG 141–034	<0.51	1.01 ± 0.20	<0.49	<0.69
IRAS 18090 + 0130	<0.30	0.67 ± 0.10	0.78 ± 0.11	0.28 ± 0.08
NGC 6701	<0.48	2.77 ± 0.22	2.26 ± 0.22	1.16 ± 0.22
NGC 6786	0.30 ± 0.10	0.51 ± 0.08	0.56 ± 0.11	0.35 ± 0.09
UGC 11415	<0.36	<0.28	0.54 ± 0.12	<0.22
ESO 593-IG 008	<0.77	<0.98	<1.22	<1.05
NGC 6907	<0.83	2.29 ± 0.31	1.27 ± 0.27	<0.65
IRAS 21101 + 5810	0.47 ± 0.11	0.53 ± 0.09	0.33 ± 0.08	<0.22
ESO 602-G 025	<0.57	0.84 ± 0.21	0.75 ± 0.12	0.74 ± 0.20
UGC 12150	0.69 ± 0.15	1.36 ± 0.17	1.59 ± 0.14	0.51 ± 0.14
IRAS F22491–1808	<0.83	<0.82	<0.73	<0.71
CGCG 453–062	0.74 ± 0.14	0.97 ± 0.16	0.48 ± 0.16	<0.42
NGC 7591	1.37 ± 0.20	1.35 ± 0.18	0.69 ± 0.13	0.68 ± 0.20
IRAS F23365 + 3604	<0.58	0.81 ± 0.27	1.28 ± 0.27	<0.88

Notes. Col 1: Source name, Col 2: $\mathrm{CCH}(N=1\to 0)$ flux with 1σ errors, Col 3: HCN ($J=1\to 0)$ flux with 1σ errors, Col 4: ${\mathrm{HCO}}^{+}(J=1\to 0)$ flux with 1σ errors, Col 5: $\mathrm{HNC}(J=1\to 0)$ flux with 1σ errors. The quoted upper limits are 3σ.

^a Though formally a detection (>3σ), this CCH flux appears to be spurious, as the signal at the location of CCH is significantly broader than would be expected.

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In order to explore the influence of AGNs on the HCN (1–0) and ${\mathrm{HCO}}^{+}(1-0)$ emission, we use the equivalent width (EQW) of the 6.2 μm polycyclic aromatic hydrocarbon (PAH) measured from Spitzer/IRS low-resolution observations (Stierwalt et al. 2013). The EQW of the 6.2 μm PAH feature compares the strength of the PAH emission associated with star formation to the strength of the mid-infrared continuum (e.g., Genzel et al. 1998; Armus et al. 2004, 2007). The AGN is energetically more important in systems with lower values of PAH EQW. Points in Figures 2–7 are color-coded by their 6.2 μm PAH EQW to show the relationship between AGNs and starburst-dominated systems.

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In addition to our new measurements, we incorporate observations from Graciá-Carpio et al. (2006), Costagliola et al. (2011), and García-Burillo et al. (2012) in our analysis. The combination of these three samples includes 63 individual galaxies with measured 6.2 μm PAH EQWs and detections in at least HCN (1–0) or ${\mathrm{HCO}}^{+}(1-0)$, comprising roughly 25% of the objects in the GOALS sample.

We use L_IR measurements from IRAS observations (Armus et al. 2009). For galaxies in pairs, we assign a portion of L_IR to each component based on their 70 or 24 μm flux ratio, as described by Diaz-Santos et al. (2013). Measurements of the [C ii] 158 μm line are taken from Diaz-Santos et al. (2013).

2.2.1. Comparison with Previous Measurements

2.3.1. Source Distance Effects

2.3.2. Spitzer versus IRAM 30 m Telescope Aperture Comparison

To consider the possible influence of XDRs resulting from powerful AGNs, we investigated the HCN/HCO⁺ ratio as a function of X-ray properties, such as the hardness ratio and the total 0.5–10 keV X-ray luminosity, from Chandra X-ray observations of the GOALS sample (Iwasawa et al. 2011). The HCN/HCO⁺ ratio showed no correlation with either the hardness ratio or the X-ray luminosity, albeit with only 10 sources common to both samples. Additional X-ray observations would be useful to further investigate the influence of XDRs. However, based on the currently available X-ray data, it does not appear that either the hardness of the X-ray spectrum or the total X-ray luminosity correlate with HCN/HCO⁺, suggesting that an XDR is not generally a major driver in enhancing HCN (1–0) for these systems, possibly because the XDRs are spatially disconnected from the regions that dominate the global line luminosity.

A complication is that enhancement from X-rays may be most effective in obscured AGNs and diagnosing this activity is difficult with observations below 10 keV. Hard X-ray observations with NuSTAR (Harrison et al. 2013), though time consuming, would provide an interesting test for the presence of obscured AGNs in sources with enhanced HCN emission.

In compact environments ${\mathrm{HCO}}^{+}(1-0)$ appears to be more susceptible to self-absorption than HCN (1–0) (Aalto et al. 2015), which would lead to an increase in the observed HCN/HCO⁺ ratio. To test if elevated ratios are primarily associated with dense systems, in Figure 5 we compare the HCN/HCO⁺ ratio with the [C ii]/L_FIR ratios from the central Herschel spaxel ($9\buildrel{\prime\prime}\over{.} 4\times 9\buildrel{\prime\prime}\over{.} 4$) as measured by Diaz-Santos et al. (2013). In local galaxies, the [C ii]/L_FIR value decreases with increasing L_FIR and dust temperature, T_dust. That the "[C ii] deficit" tracks T_dust suggests the deficit is the result of compact nuclear activity. Diaz-Santos et al. (2013) find that this trend of a lower [C ii]/L_FIR ratio with decreasing source size is present when considering only starbursts, suggesting that the [C ii] deficit is driven by the compactness of the starburst, rather than dust heated by an AGN.

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Therefore, if the HCN (1–0)/${\mathrm{HCO}}^{+}(1-0)$ ratio is being driven primarily by the density/compactness of the starburst, we should see a correlation between the two quantities. For sources with [C ii]/L_FIR > 10⁻³, the signal-to-noise weighted mean ${L}_{\mathrm{HCN}(1-0)}^{\prime }$ to ${L}_{{\mathrm{HCO}}^{+}(1-0)}^{\prime }$ is 1.0 ± 0.4, while sources with [C ii]/L_FIR < 10⁻³ (large [C ii] deficits) have a mean ratio of 1.7 ± 0.5. Formally, the ratio does not appear to vary as a function of [C ii]/L_FIR, though a Spearman rank analysis suggests a moderate anti-correlation (ρ = − 0.401 ± 0.142, z-score = − 1.165 ± 0.458; Table 3). This is somewhat weaker than the anti-correlation of HCN/HCO⁺ with PAH EQW. The majority of the sources with [C ii]/L_FIR < 10⁻³ have HCN/HCO⁺ ratios >1, consistent with a scenario in which a compact and dense starburst causes an enhancement of HCN (1–0). On the other hand, a substantial number of systems without significant [C ii]/L_FIR deficits also have HCN/HCO⁺ > 1. For composite and starburst systems (PAH EQW > 0.2 μm) higher HCN/HCO⁺ ratios do not appear to be associated with lower [C ii]/L_FIR values. Based on the available data, we cannot directly link the [C ii] deficit with enhanced HCN (1–0) emission.

It is worth noting that HCN (1–0) can be enhanced in sources that do not show a strong [C ii] deficit; if a compact continuum source is surrounded by extended disk-like star formation, the system may have a "normal" [C ii]/L_FIR ratio, but still show enhanced HCN (1–0) associated with the compact starburst on scales that cannot be resolved by Herschel. See Diaz-Santos et al. (2014) for a discussion on extended [C ii] emission in the GOALS sample. Assessing the spatial distribution of the HCN (1–0)/${\mathrm{HCO}}^{+}(1-0)$ would be a useful comparison with [C ii]/L_FIR—do regions with "normal" [C ii]/L_FIR correspond to regions with low HCN (1–0)/${\mathrm{HCO}}^{+}(1-0)$?

A possible concern with the above analysis is that the L_IR, [C ii]/L_FIR, and the PAH EQW are all correlated. Indeed, the L_IR, the [C ii] deficit, and the 6.2 μm PAH EQW are interrelated in that the most luminous sources tend to have the strongest [C ii] deficits (Diaz-Santos et al. 2013, because both L_IR and [C ii]/L_FIR correlate with the dust temperature;) and lower PAH EQWs (Petric et al. 2011; Stierwalt et al. 2013). However, the converse is not true; a normal [C ii]/L_FIR value does not guarantee a high PAH EQW. Systems with low or intermediate PAH EQWs are distributed across the range of L_IR and [C ii]/L_FIR values seen for our sample (e.g., Figure 5 and Diaz-Santos et al. 2013). In contrast, essentially all of the high PAH EQW (starburst) systems have normal [C ii]/L_FIR and L_IR∼ a few ×10¹¹ L_⊙. Thus, while these quantities are generally related, systems with low or intermediate PAH EQWs appear to have a range of compactness (as diagnosed by [C ii]/L_FIR), so the PAH EQW and [C ii] deficit appear to be semi-independent. That the HCN (1–0)/${\mathrm{HCO}}^{+}(1-0)$ ratio does not correlate well with either suggests that the origin of HCN enhancement cannot be easily assigned solely to an AGN or to the presence of a compact starburst.

The strong mid-infrared continuum present in many (U)LIRGs may also influence these line ratios. The HCN molecule has degenerate bending modes in the infrared and can absorb mid-infrared photons (14 μm) to its first vibrational state. The transitions have high level energies (>1000 K) and mid-IR emission with a brightness temperature of at least 100 K is necessary to excite them (Aalto et al. 2007). A sign that infrared pumping of HCN is taking place is the presence of line emission from rotational transitions within the vibrational band (e.g., Sakamoto et al. 2010). However, it may be possible for HCN to be infrared pumped without detectable emission from these rotational-vibrational lines. When the brightness temperature is close to the lower limit for pumping, the excitation of the rotational-vibrational line is so low that it takes a very large column density to result in a large enough optical depth for this rotational-vibrational line to be detected.

Several sources in our sample do show evidence of infrared absorption at 14 μm, which could be associated with the pumping of HCN (Lahuis et al. 2007; Sakamoto et al. 2010), but none of the starburst systems with enhanced HCN emission show 14 μm absorption in the Spitzer/IRS observations from Inami et al. (2013).

HCO⁺ is similarly susceptible to infrared pumping, via a ∼12 μm line; using the IRS high-resolution spectroscopy from Inami et al. (2013) we searched for a ∼12 μm absorption feature in those systems with high ratios, where we postulated mid-infrared pumping could be important. We did not find any evidence for HCO⁺ absorption.

However, it is not clear that existing mid-infrared observations are sensitive enough to establish firm upper limits on the importance of infrared pumping. Existing spectroscopy may not have the sensitivity to identify absorption that is sufficient to affect level populations.

An obvious question is whether or not the PAH EQW is a robust measure of AGN strength in (U)LIRGs. Aalto et al. (2015) present interferometric observations of several objects, including the composite source CGCG 049–057, one of our sources with enhanced HCN emission. The HCN (3–2) v₂ = 0 and ${\mathrm{HCO}}^{+}(3-2)$ lines show evidence for significant self-absorption and the HCN (3–2) v₂ = 1 ro-vibrational line is detected (consistent with mid-infrared pumping). Aalto et al. (2015) interpret this as evidence of a very compact (tens of pc) warm (>100 K) region surrounded by a large envelope of cooler, more diffuse gas. The total column density through these systems is likely quite high (>10²⁴ cm⁻²). If the column densities are sufficiently high to support mid-infrared pumping, the optical depths in the infrared may be high enough that the mid-infrared continuum level does not trace the intrinsic energy source—thus the PAH EQW diagnostic may miss highly embedded AGNs.

In Section 2.3.2 we note that the PAH EQW is an upper limit to the AGN's influence on large scales, but this measurement could also be considered as a lower limit to the AGN's influence on smaller scales. Thus, consistent with results of Aalto et al. (2015) for CGCG 047–059, it is possible that these starburst or composite sources with high HCN (1–0)/${\mathrm{HCO}}^{+}(1-0)$ ratios host embedded AGNs, and substantial mid-infrared optical depths result in an underestimate of the AGN's influence when using the PAH EQW.

Though we have shown in this study that some starburst-dominated systems have global ${L}_{\mathrm{HCN}(1-0)}^{\prime }$/${L}_{{\mathrm{HCO}}^{+}(1-0)}^{\prime }$ ratios in excess of unity, the starburst-dominated systems outnumber AGN-dominated systems in the present sample. More IRAM 30 m single-dish observations of AGN-dominated (U)LIRGs would be useful to study the dense-gas tracers in that population, particularly to see if the absence of AGN-dominated sources with low ${L}_{\mathrm{HCN}(1-0)}^{\prime }$/${L}_{{\mathrm{HCO}}^{+}(1-0)}^{\prime }$ ratios is truly representative or merely reflects small number statistics.

Spatially resolved comparisons of the HCN emission with CO or HCO⁺ have been undertaken for systems known to host an AGN (e.g., Kohno et al. 2003; Imanishi et al. 2006, 2007, 2009; Davies et al. 2012; Imanishi & Nakanishi 2013, 2014); it would be instructive to perform the same exercise on starburst-dominated systems. The Atacama Large Millimeter/sub-millimeter Array is the natural instrument for this, and such a study could investigate the spatial variation in the ${L}_{\mathrm{HCN}(1-0)}^{\prime }$/${L}_{{\mathrm{HCO}}^{+}(1-0)}^{\prime }$ ratio for starbursts. If the ratio peaks on the nucleus but is low elsewhere (as in systems with AGNs), does that correspond to a compact mid-infrared emitting region? Spatially resolving this ratio in starburst systems with both high and low ratios may enable a separation of the relative importance of source compactness and/or mid-infrared pumping in setting the ratio.

Krips et al. (2008) studied the emission of multiple HCN and HCO⁺ transitions for 12 systems, finding evidence for systematic differences in the HCN (1–0)/${\mathrm{HCO}}^{+}(1-0)$ and HCN (3–2)/${\mathrm{HCO}}^{+}(3-2)$ ratios between AGNs and starbursts, with AGNs having values >2 for both ratios. They also found evidence that the (3–2)/(1–0) ratio for both HCN and HCO⁺ varies between AGNs and starbursts. The number of GOALS objects with (3–2) measurements of HCN and HCO⁺ is currently too small to draw conclusions regarding the (3–2)/(1–0) ratios as a function of AGN strength, but it would be possibly illuminating to compare the higher J_upper transitions with the AGN diagnostic used here.

The discovery of mid-infrared pumping and associated high-column densities in enhanced HCN sources suggests that even mid-infrared diagnostics such as the PAH EQW may miss highly embedded AGNs. If this is the case, hard X-ray observations may provide the only conclusive evidence for AGNs in sources with heavily obscured nuclei. However, the weakness of the >10 keV X-rays in ULIRG AGNs (e.g., Mrk 231; Teng et al. 2014) may make these observations difficult with existing facilities.

Finally, the influence of infrared pumping is still uncertain. Future mid-infrared observations of these sources with the James Webb Space Telescope could provide tighter constraints on the importance and ubiquity of mid-infrared pumping, for both HCN and HCO⁺. See also Aalto et al. (2007) for a discussion on using resolved observations to distinguish mid-infrared pumping and XDR scenarios.

We now turn to the relationship between the HCN (1–0) emission and the SFRs in (U)LIRGs. While HCN (1–0) is generally taken to trace the mass of dense molecular gas, M_dense, previous studies have found conflicting results for the relationship between L_IR and ${L}_{\mathrm{HCN}(1-0)}^{\prime }$. In Figure 6 (left) we plot the standard relation between ${L}_{\mathrm{HCN}(1-0)}^{\prime }$ and L_IR including additional measurements from Gao & Solomon (2004b, with values corrected to our assumed cosmology and for the 3-attractor model of Mould et al. 2000).

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We fit the relationship using a maximum likelihood technique, considering the errors in both L_IR and ${L}_{\mathrm{HCN}(1-0)}^{\prime }$ (e.g., Section 7 of Hogg et al. 2010). When uncertainties for millimeter-line fluxes were not quoted in the literature, we assume 20%. Uncertainties on the fit parameters were determined using Markov Chain Monte Carlo (MCMC) sampling ²⁰ and quoted fit parameter uncertainties are for the 99% confidence interval.

To this point, we have not considered systematic uncertainties in our new observations, as the simultaneous measurement of HCN and HCO⁺ should result in significantly reduced systematic uncertainties in the HCN/HCO⁺ ratio, in the HCN/HCO⁺ ratio, compared to non-simultaneous measurements. However, for the present comparison of L_IR with these lines, the absolute flux of these lines may be subject to systematic uncertainties. We tested two approaches for dealing with systematic uncertainties: (1) assigning additional systematic uncertainties (equal in magnitude to the statistical uncertainties) to each data point ²¹ and fitting the relationship, or (2) including only statistical uncertainties ²² and allowing for additional uncertainties in the fitting process. In addition to fitting for the slope and intercept, we included an additional "nuisance" parameter, f, in the likelihood function to fit for the fractional amount by which the uncertainties are underestimated (e.g., due to not explicitly including systematic uncertainties). After the MCMC sampling, we marginalized over f when determining the final uncertainties for the slope and intercept. Both approaches yielded the same results for the best-fit relations, so we conclude that our fitting process appropriately accounts for non-statistical uncertainties and present the numerical results from the latter approach. For consistency, we show only the statistical uncertainties for values plotted in Figures 6 and 7.

Our best-fit relation to our new data combined with the literature data (omitting both AGN-dominated systems, in which L_IR may not solely trace star formation, and HCN (1–0) upper limits) between L_IR and ${L}_{\mathrm{HCN}(1-0)}^{\prime }$ is:

$$\begin{eqnarray}&&{\mathrm{log}}_{10}{L}_{\mathrm{IR}}=({1.08}_{-0.16}^{+0.18}){\mathrm{log}}_{10}{L}_{\mathrm{HCN}(1-0)}^{\prime }+({1.47}_{-0.97}^{+1.00}).\end{eqnarray} \tag{ 1 }$$

This fit is shown in Figure 6 (left) as the solid line. We also fit ${L}_{\mathrm{HCN}(1-0)}^{\prime }$–L_IR, fixing the slope to be linear, which we show in Figure 6 (left) as a dashed line. The slope of the ${L}_{\mathrm{HCN}(1-0)}^{\prime }$–L_IR relation is consistent with being linear at the ∼1.3σ level. We note that García-Burillo et al. (2012) found the ${L}_{\mathrm{HCN}(1-0)}^{\prime }$–L_FIR[40 − 400 μm] relation to be steeper than linear, with a slope of 1.23 ± 0.05 (68% confidence interval).

In Figure 6 (right) we perform the same comparison, but utilizing ${L}_{{\mathrm{HCO}}^{+}(1-0)}^{\prime }$ instead of ${L}_{\mathrm{HCN}(1-0)}^{\prime }$. As ${\mathrm{HCO}}^{+}(1-0)$ has a higher critical density than CO (1–0), but lower than HCN (1–0), it is useful to test if it is also linearly tracked by the SFR. The ${L}_{{\mathrm{HCO}}^{+}(1-0)}^{\prime }$–L_IR relation is also consistent with being linear:

$$\begin{eqnarray}&&{\mathrm{log}}_{10}{L}_{\mathrm{IR}}=({0.94}_{-0.19}^{+0.25}){\mathrm{log}}_{10}{L}_{{\mathrm{HCO}}^{+}(1-0)}^{\prime }+({2.73}_{-1.75}^{+1.63}).\end{eqnarray} \tag{ 2 }$$

Thus, our results are consistent with scenarios in which the HCN and HCO⁺ emission both trace the dense gas associated with ongoing star formation, albeit with scatter, likely the result of one or more of the excitation mechanisms discussed in Section 4.

The inclusion of the Gao & Solomon (2004a) HCN observations adds in a substantial number of galaxies with L_IR < 10¹¹ L_⊙, but corresponding ${\mathrm{HCO}}^{+}(1-0)$ observations are not available. In order to assess the degree to which these lower-luminosity systems influence our fitting results, we also fit the HCN (1–0)–L_IR relation without the points from Gao & Solomon (2004a); this fit is shown as the dotted–dashed line in Figure 7 (Left) and the best-fit relation is

$$\begin{eqnarray}&&{\mathrm{log}}_{10}{L}_{\mathrm{IR}}=({1.09}_{-0.31}^{+0.48}){\mathrm{log}}_{10}{L}_{\mathrm{HCN}(1-0)}^{\prime }+({1.40}_{-2.27}^{+2.08}),\end{eqnarray} \tag{ 3 }$$

which is consistent with that for the full data set.

The ratio ${L}_{\mathrm{HCN}(1-0)}^{\prime }$/L_IR is often taken as ∝M_dense/SFR, which corresponds to the depletion time (τ_dep) of the dense molecular gas. In Figure 7 (left) we show ${L}_{\mathrm{HCN}(1-0)}^{\prime }$/L_IR as a function of L_IR, excluding sources that are AGN-dominated (PAH EQW < 0.2 μm). As in Figure 6 (left), we include observations from Gao & Solomon (2004a). The mean log₁₀(${L}_{\mathrm{HCN}(1-0)}^{\prime }$/L_IR) = 1 × 10⁻³ with an rms scatter of 0.22 dex. The AGN- and starburst-dominated systems (as traced by the 6.2 μm PAH feature) have similar ${L}_{\mathrm{HCN}(1-0)}^{\prime }$/L_IR ratios.

Similarly, ${L}_{{\mathrm{HCO}}^{+}(1-0)}^{\prime }$/L_IR shows no obvious trend with L_IR, but has substantial scatter. The mean log₁₀(${L}_{{\mathrm{HCO}}^{+}(1-0)}^{\prime }$/L_IR) = 1 × 10⁻³, with an rms scatter of 0.19 dex.

The significant scatter in ${L}_{\mathrm{HCN}(1-0)}^{\prime }$/${L}_{{\mathrm{HCO}}^{+}(1-0)}^{\prime }$ can plausibly be attributed to the influence of multiple excitation mechanisms for these dense-gas tracers, as described above. These observations are consistent with a scenario in which molecular abundance (Lepp & Dalgarno 1996), density (Meijerink et al. 2007), and radiative effects (Aalto et al. 1995) all influence the global HCN (1–0) emission, obscuring a simple link between ${L}_{\mathrm{HCN}(1-0)}^{\prime }$ and the mass of dense gas directly associated with ongoing star formation. Stated differently, the conversion factor between ${L}_{\mathrm{HCN}(1-0)}^{\prime }$ (or ${L}_{{\mathrm{HCO}}^{+}(1-0)}^{\prime }$) and M_dense likely depends on the relative HCN–H₂ abundance in addition to the overall density, excitation source (PDR versus XDR), and influence of infrared pumping. A better understanding of the processes in (U)LIRGs contributing to ${L}_{\mathrm{HCN}(1-0)}^{\prime }$ and ${L}_{{\mathrm{HCO}}^{+}(1-0)}^{\prime }$ are needed to determine if the scatter in Figure 7 is due to differences in the consumption rates of molecular gas or a varying ${L}_{\mathrm{HCN}(1-0)}^{\prime }$–M_dense conversion factor. Future observations (Section 4.5), coupled with improved modeling, should be able to discriminate between these scenarios for the HCN and HCO⁺ emission in (U)LIRGs.