SUBMILLIMETER-HCN DIAGRAM FOR ENERGY DIAGNOSTICS IN THE CENTERS OF GALAXIES

In our RADEX simulation, we investigated how the following parameters affect the line ratios of our interest.

• 1.  
  Kinetic temperature (T_kin) of the molecular gas: this affects the rate of the collisional excitation with the target molecules. The cases of 50, 100, and 200 K are investigated. This range mostly covers the T_kin suggested for nearby AGNs and SB galaxies (e.g., Mauersberger et al. 2003; Krips et al. 2008; Davies et al. 2012; Viti et al. 2014).
• 2.  
  Molecular gas density (${n}_{{{\rm{H}}}_{2}}$): this also affects the rate of collisional excitation. Two cases of ${n}_{{{\rm{H}}}_{2}}$ = 10⁵ and 5 × 10⁶ cm⁻³ will be examined. These values are typical ones suggested in nuclear regions of galaxies (e.g., I13; Viti et al. 2014), which is also supported by the commonly subthermal excitation of our target lines (e.g., I13; Knudsen et al. 2007; Viti et al. 2014).
• 3.  
  Ratios of molecular fractional abundances with respect to H₂ (X_mol, mol = HCN, HCO⁺, and CS): we will show two cases for simplicity, where ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$ (or ${X}_{\mathrm{HCN}}/{X}_{\mathrm{CS}}$) = 1 and 10.
• 4.  
  Background radiation temperature (T_bg): molecular rotational levels can be radiatively excited through absorbing photons. In this perspective, it is highly likely that the background radiation is stronger around AGNs than in SB environments. In fact, the dust temperature at far-IR to submillimeter (i.e., Rayleigh–Jeans regime) is as high as 46 K in the central ∼400 pc region of NGC 1068 (AGN; García-Burillo et al. 2014), but 29 K in the central 1.2 kpc region of NGC 253 (SB; Weiß et al. 2008), for example. The cases of 2.73 (cosmic microwave background), 5, 10, 20, 30, 40, 50, and 60 K are studied. We simply adopt the blackbody approximation to calculate the background radiation field.
• 5.  
  Optical depth of the line emission (τ): models with different N_mol/dV (or equivalently a volume density of the target molecule to a velocity gradient ratio) are employed to test this effect. Here N_mol and dV are the line-of-sight column density and line velocity width, respectively. ${N}_{{\mathrm{HCO}}^{+}}$/dV (or N_CS/dV) = 5.0 × 10¹², 5.0 × 10¹³, 5.0 × 10¹⁴, and 5.0 × 10¹⁵ cm⁻² (km s⁻¹)⁻¹ are studied. The N_HCN/dV is equated to the above ${N}_{{\mathrm{HCO}}^{+}}$/dV (or N_CS/dV), or enhanced by 10 times, since X_HCN/${X}_{{\mathrm{HCO}}^{+}}$ (or X_HCN/X_CS) = 1 and 10 as already mentioned.

Under these conditions, we ran RADEX for each set of (T_kin, ${n}_{{{\rm{H}}}_{2}}$, X_HCN/${X}_{{\mathrm{HCO}}^{+}}$, T_bg, ${N}_{{\mathrm{HCO}}^{+}}$/dV) or (T_kin, ${n}_{{{\rm{H}}}_{2}}$, X_HCN/X_CS, T_bg, N_CS/dV) and took line ratios of ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ and ${R}_{\mathrm{HCN}/\mathrm{CS}}$. Transitions between vibrational levels through IR pumping are not included in the models since there is no detection of vibrationally excited HCN emission in our sample galaxies except for NGC 4418 (Sakamoto et al. 2010), although it can be a bit of an inappropriate treatment in some cases (Section 5). Regarding the gas density, we here mention that Krips et al. (2008) suggested lower ${n}_{{{\rm{H}}}_{2}}$ in AGNs than in SB galaxies, but their measurements were based on single-dish observations of, e.g., the HCN(3–2)-to-HCN(1–0) line ratio. On the other hand, we need spatially resolved measurements of line ratios to accurately assess ${n}_{{{\rm{H}}}_{2}}$ or line excitation (Viti et al. 2014). One may expect that an increase in ${n}_{{{\rm{H}}}_{2}}$ naturally leads to higher line ratios since n_cr of HCN(4–3) is the highest among the target lines. However, this is actually not so straightforward, as shown later.

In the modeling below, we first fix ${n}_{{{\rm{H}}}_{2}}$ to be 10⁵ cm⁻³ and investigate the dependence of the line ratios on the other parameters. The case of ${n}_{{{\rm{H}}}_{2}}$ = 5 × 10⁶ cm⁻³ will be shown subsequently. Note that our modeling is fundamentally different from the LTE modelings by, e.g., I13, Viti et al. (2014), and Martín et al. (2015), in the sense that non-LTE processes are treated here. Moreover, we also examine the dependence of the line ratios on T_bg, which has not been studied in the non-LTE modelings by, e.g., Krips et al. (2011) and Viti et al. (2014). Hence, in addition to the previous key works, we expect that our analysis will provide some insight on the underlying physical and chemical conditions in the centers of galaxies.

Excitation states of the target lines under the fixed gas density of ${n}_{{{\rm{H}}}_{2}}$ = 10⁵ cm⁻³ are described here. HCN(4–3) is mainly used as a representative case, but the same argument can hold for HCO⁺(4–3) and CS(7–6) as well. Figure 3 shows excitation temperature (T_ex) of HCN(4–3) as a function of T_bg. When the line emission is optically thin or moderately thick (panels (a) and (b) in Figure 3), one can find that the excitation is dominated by radiative processes, since T_ex is very close to T_bg especially at T_bg ≳ 10 K. Note that ${\rm{\Delta }}{E}_{\mathrm{ul}}$ of HCN(4–3) is 17.0 K (Table 2). The excitation states of HCO⁺(4–3) and CS(7–6) are presented in Appendix B. The optical depths of the target lines calculated by RADEX are shown in Figure 4 as a function of T_bg.

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The excitation state shown in Figure 3 can be better understood through a two-level (e.g., HCN J = 3 and 4) analytic treatment. Although we conducted a full-level statistical equilibrium calculation with RADEX here, this analytic treatment provides us fruitful insights on what is influencing the molecular excitation. A similar approach can be found in, e.g., Scoville et al. (2015) and Scoville & Solomon (1974), but we include background radiation in the analysis as well, which has usually been omitted.

The ratio of the upper- to lower-level molecular population with the energy gap of ${\rm{\Delta }}{E}_{{ul}}$ can be written as

$$\begin{eqnarray}\displaystyle \frac{{n}_{u}}{{n}_{l}} & = & \displaystyle \frac{{B}_{{lu}}{J}_{\nu }+{C}_{{lu}}}{{A}_{{ul}}+{B}_{{ul}}{J}_{\nu }+{C}_{{ul}}}=\displaystyle \frac{{g}_{u}}{{g}_{l}}\displaystyle \frac{\eta {A}_{{ul}}{\beta }_{\nu }+{C}_{{ul}}\mathrm{exp}\left(-\displaystyle \frac{{\rm{\Delta }}{E}_{{ul}}}{{T}_{\mathrm{kin}}}\right)}{(1+\eta ){A}_{{ul}}{\beta }_{\nu }+{C}_{{ul}}}\\ & = & \displaystyle \frac{{g}_{u}}{{g}_{l}}\mathrm{exp}\left(-\displaystyle \frac{{\rm{\Delta }}{E}_{{ul}}}{{T}_{\mathrm{ex}}}\right),\end{eqnarray} \tag{ 1 }$$

with

$$\begin{eqnarray}&&\eta \equiv \displaystyle \frac{1}{\mathrm{exp}\left(\displaystyle \frac{{\rm{\Delta }}{E}_{{ul}}}{{T}_{\mathrm{bg}}}\right)-1}.\end{eqnarray} \tag{ 2 }$$

Here A_ul, B_ul, and C_ul indicate the Einstein coefficients of spontaneous decay, stimulated emission, and collisional de-excitation, respectively. The upward and downward collision rates are related assuming a detailed balance in the thermodynamic equilibrium state. The g_u and g_l are statistical weights of the upper and lower levels, respectively. The frequency of the line is represented as ν. The internal radiation J_ν is

$$\begin{eqnarray}&&{J}_{\nu }=(1-{\beta }_{\nu }){B}_{\nu }({T}_{\mathrm{ex}})+{\beta }_{\nu }{B}_{\nu }({T}_{\mathrm{bg}}),\end{eqnarray} \tag{ 3 }$$

where ${B}_{\nu }(T)$ is the Planck function at temperature T. Although we should include various mechanisms as the source of the background radiation and solve their radiative transfer individually to achieve the local spectral energy distribution, we represent them by a single Planck function throughout this paper for simplicity. The photon escape probability from the model cloud is denoted as β_ν, which is a function of the line optical depth τ as

$$\begin{eqnarray}&&{\beta }_{\nu }=\displaystyle \frac{1.5}{\tau }\left[1-\displaystyle \frac{2}{{\tau }^{2}}+\left(\displaystyle \frac{2}{\tau }+\displaystyle \frac{2}{{\tau }^{2}}\right){e}^{-\tau }\right],\end{eqnarray} \tag{ 4 }$$

for a spherical cloud (e.g., Osterbrock & Ferland 2006). The value of β_ν also depends on the assumed geometry. In Equation (1), A_ul is reduced by β_ν, which means that an effective n_cr of the line (${n}_{\mathrm{cr},\mathrm{eff}}$) is lower than the n_cr at the optically thin limit (${n}_{\mathrm{cr},\mathrm{thin}};$ Table 2) owing to the photon trapping effect, i.e., ${n}_{\mathrm{cr},\mathrm{eff}}$ = ${\beta }_{\nu }\;\times \;{n}_{\mathrm{cr},\mathrm{thin}}$. The background radiation field is included in Equation (1) as η. We show the η of HCN(4–3) in Figure 5, which is almost identical to those of HCO⁺(4–3) and CS(7–6).

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By introducing ${n}_{\mathrm{cr},\mathrm{eff}}$, Equation (1) is reduced to

$$\begin{eqnarray}\mathrm{exp}\left(-\displaystyle \frac{{\rm{\Delta }}{E}_{{ul}}}{{T}_{\mathrm{ex}}}\right) & = & \displaystyle \frac{\eta +\left(\displaystyle \frac{{n}_{{{\rm{H}}}_{2}}}{{n}_{\mathrm{cr},\mathrm{eff}}}\right)\mathrm{exp}\left(-\displaystyle \frac{{\rm{\Delta }}{E}_{{ul}}}{{T}_{\mathrm{kin}}}\right)}{(1+\eta )+\left(\displaystyle \frac{{n}_{{{\rm{H}}}_{2}}}{{n}_{\mathrm{cr},\mathrm{eff}}}\right)}\\ & & \sim \displaystyle \frac{\eta +\left(\displaystyle \frac{{n}_{{{\rm{H}}}_{2}}}{{n}_{\mathrm{cr},\mathrm{eff}}}\right)}{(1+\eta )+\left(\displaystyle \frac{{n}_{{{\rm{H}}}_{2}}}{{n}_{\mathrm{cr},\mathrm{eff}}}\right)}.\end{eqnarray} \tag{ 5 }$$

Here exp(−${\rm{\Delta }}{E}_{{ul}}/{T}_{\mathrm{kin}}$) commonly takes ∼1 for HCN(4–3), HCO⁺(4–3), and CS(7–6) at any T_kin in our models. With this formula, the molecular excitation in the two extreme cases of β_ν, i.e., ${\beta }_{\nu }\to 1$ (optically thin limit) and ${\beta }_{\nu }\to 0$ (optically thick limit), is discussed in the following.

(i) Optically thin limit—in this limit, we expect ${n}_{{{\rm{H}}}_{2}}/{n}_{\mathrm{cr},\mathrm{eff}}$ = ${n}_{{{\rm{H}}}_{2}}/{n}_{\mathrm{cr},\mathrm{thin}}\;\sim$ 10⁻² for HCN(4–3), for example (Table 2). Taking Figure 5 into account as well, one can find that both collisional and radiative excitation can contribute to the molecular excitation, especially when T_bg ≲ 5 K. In this range, $\eta \lesssim {n}_{{{\rm{H}}}_{2}}/{n}_{\mathrm{cr},\mathrm{eff}}$. On the other hand, the right-hand side of Equation (5) reduces to $\mathrm{exp}(-{\rm{\Delta }}{E}_{{ul}}/{T}_{\mathrm{bg}})$, i.e., molecules are radiatively excited. This trend stands out especially when T_bg ≳ 10 K, which can be clearly seen in Figure 3(a). Therefore, as a general manner, we suggest that radiative excitation should be considered seriously when we treat optically thin to moderately thick line emissions from AGNs, where high T_bg is likely expectable.

(ii) Optically thick limit—in this limit, by substituting ${\beta }_{\nu }\;\to$ 0 (or ${n}_{\mathrm{cr},\mathrm{eff}}\;\to$ 0) into Equation (5), one can find that T_ex is now identical to T_kin and thus independent of T_bg. Indeed, T_ex is less dependent on T_bg and also differs a lot according to T_kin (panels (c) and (d) in Figure 3), as HCN(4–3) becomes optically thicker (see also Figure 4). We note that the output parameters have a limited meaning at a high optical depth such as τ ≳ 100 in the RADEX code, since the change of optical depth over the line profile is not taken into account (van der Tak et al. 2007).

We then calculated ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ and ${R}_{\mathrm{HCN}/\mathrm{CS}}$ based on our non-LTE modelings as a function of T_bg. Several cases with different T_kin, N_mol/dV (or τ), and molecular abundance ratios are shown in Figure 6. Here we discuss a dependence of each line ratio on the parameters in our models. We guide readers to Martín et al. (2015) for these line ratios calculated under the LTE condition.

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(i) ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$—one will find in Figures 6(a)–(d) that this ratio is not so sensitive to T_bg, which is close to T_ex when the excitation is dominated by the radiative processes (see also Figure 3). To further examine this trend, we rewrite ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ analytically as

$$\begin{eqnarray}&&\displaystyle \frac{{T}_{\mathrm{ex},\mathrm{HCN}(4-3)}-{T}_{\mathrm{bg}}}{{T}_{\mathrm{ex},{\mathrm{HCO}}^{+}(4-3)}-{T}_{\mathrm{bg}}}\cdot \displaystyle \frac{1-\mathrm{exp}({-\tau }_{\mathrm{HCN}(4-3)})}{1-\mathrm{exp}({-\tau }_{{\mathrm{HCO}}^{+}(4-3)})}\\ &&\quad \equiv {\xi }_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}\cdot {\zeta }_{\mathrm{HCN}/{\mathrm{HCO}}^{+}},\end{eqnarray} \tag{ 6 }$$

where ${\xi }_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ and ${\zeta }_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ correspond to the former and the latter term of the left-hand side of Equation (6), respectively. Also, we rewrite the optical depth as

$$\begin{eqnarray}&&{\tau }_{{ul}}=\displaystyle \frac{{c}^{3}}{8\pi {\nu }^{3}}\displaystyle \frac{{g}_{u}}{{g}_{l}}{A}_{{ul}}\displaystyle \frac{{N}_{l}}{{dV}}\left(1-\mathrm{exp}\left(-\displaystyle \frac{{\rm{\Delta }}{E}_{{ul}}}{{T}_{\mathrm{ex}}}\right)\right),\end{eqnarray} \tag{ 7 }$$

where N_l is the line-of-sight column density at the lower energy level. The resultant ${\xi }_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ and ${\zeta }_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ are plotted in Figure 7(a) for the representative case of ${N}_{{\mathrm{HCO}}^{+}}/{dV}$= 5 × 10¹³ cm⁻² (km s⁻¹)⁻¹. At this ${N}_{{\mathrm{HCO}}^{+}}/{dV}$ with ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$ = 1, HCN(4–3) and HCO⁺(4–3) are moderately optically thick, whereas HCN(4–3) can be heavily optically thick when ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$=10 (see also Figure 4). From Figure 7(a), one may find that ${\xi }_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ and ${\zeta }_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ vary toward the opposite direction at ${T}_{\mathrm{bg}}\;\lesssim$ 10 K (the range where both collisional and radiative processes can influence the excitation), which compensates with each other to keep the ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ more or less constant. At T_bg ≳ 10 K where T_ex ∼ T_bg, on the other hand, both ${\xi }_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ and ${\zeta }_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ themselves are not so sensitive to T_bg anymore. Note that we can expect ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}\;\sim$ 1 when both lines are thermalized and at the optically thick limit.

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As for the dependence on T_kin, one can see that ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ increases as T_kin gets higher in each panel, reflecting the higher n_cr of HCN(4–3) than HCO⁺(4–3), except for the case of X_HCN = ${X}_{{\mathrm{HCO}}^{+}}$ in Figure 6(d). In that exceptional case, where both lines are quite optically thick, HCO⁺(4–3) is fully thermalized because of the well-reduced ${n}_{\mathrm{cr},\mathrm{eff}}$ ($\ll {n}_{{{\rm{H}}}_{2}}$ = 10⁵ cm⁻³) due to the photon trapping (see also Appendix B), whereas HCN(4–3) is still subthermally excited, and ${T}_{\mathrm{ex}}/{T}_{\mathrm{kin}}$ of HCN(4–3) is maximized at T_kin = 50 K.

Regarding the molecular abundance ratio, we find that ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}\;\gtrsim$ 10 is necessary to reproduce ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}\;\gt$ 1 observed in AGNs in any T_kin and N_mol/dV studied here. Enhancing X_HCN (equivalently enhancing N_HCN in our models) will increase ${\tau }_{\mathrm{HCN}(4-3)}$ and T_ex (photon trapping), both of which will result in subsequent enhancement of the ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$. The required ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$ is consistent with our previous multi-transitional non-LTE modeling of HCN and HCO⁺ in NGC 1097 (I13). We should also mention that Yamada et al. (2007) concluded, based on their three-dimensional radiative transfer simulations, that X_HCN must be an order of magnitude higher than ${X}_{{\mathrm{HCO}}^{+}}$ in order to account for the observed high HCN(1–0)/HCO⁺(1–0) ratios in AGNs (e.g., ∼2 in NGC 1068; Kohno et al. 2008). Our results seem to be consistent with their modelings, although we here use J = 4–3 transitions. Moreover and importantly, the ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$ required for AGNs (≳10) are significantly higher than that required to reproduce the ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ in SB galaxies, which is typically ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}\;\sim$ 3. Therefore, the boosting factor of the abundance ratio in AGNs over that in SB galaxies is at least ∼3. This factor can even increase to ≳10 (i.e., ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}\;\gtrsim$ 30) to account for the high-end values observed in AGNs (e.g., NGC 1068 (W-knot) in Table 1) based on our modelings. For a convenient discussion, we here define the boosting factor as

$$\begin{eqnarray}&&{\mathrm{BF}}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}=\displaystyle \frac{{({X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}})}_{\mathrm{AGN}}}{{({X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}})}_{\mathrm{SB}}},\end{eqnarray} \tag{ 8 }$$

where (X_HCN/${X}_{{\mathrm{HCO}}^{+}}$)_AGN and (X_HCN/${X}_{{\mathrm{HCO}}^{+}}$)_SB denote the molecular fractional abundance ratios in AGN and SB galaxies, respectively. The same notation is also used for the case of X_HCN/X_CS. Note that, for an extreme case like NGC 1068 (W-knot), we require ${N}_{{\mathrm{HCO}}^{+}}$/dV ≲ 5 × 10¹³ cm⁻² (km s⁻¹)⁻¹ because ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ will eventually converge to unity for larger optical depths and never reaches such high observed values as ∼3.

(ii) ${R}_{\mathrm{HCN}/\mathrm{CS}}$—contrary to ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$, ${R}_{\mathrm{HCN}/\mathrm{CS}}$ shows a steep dive in Figures 6(e) and (f) as T_bg increases from 2.73 to 10 K, when ${X}_{\mathrm{HCN}}/{X}_{\mathrm{CS}}$ = 10. This feature is more prominent at lower T_kin. All other cases show almost constant ${R}_{\mathrm{HCN}/\mathrm{CS}}$ against T_bg. Following the same manner in Equation (6), we express the ${R}_{\mathrm{HCN}/\mathrm{CS}}$ as

$$\begin{eqnarray}&&\displaystyle \frac{{T}_{\mathrm{ex},\mathrm{HCN}(4-3)}-{T}_{\mathrm{bg}}}{{T}_{\mathrm{ex},\mathrm{CS}(7-6)}-{T}_{\mathrm{bg}}}\cdot \displaystyle \frac{1-\mathrm{exp}({-\tau }_{\mathrm{HCN}(4-3)})}{1-\mathrm{exp}({-\tau }_{\mathrm{CS}(7-6)})}\\ &&\quad \equiv {\xi }_{\mathrm{HCN}/\mathrm{CS}}\cdot {\zeta }_{\mathrm{HCN}/\mathrm{CS}}.\end{eqnarray} \tag{ 9 }$$

The resultant ${\xi }_{\mathrm{HCN}/\mathrm{CS}}$ and ${\zeta }_{\mathrm{HCN}/\mathrm{CS}}$ are plotted in Figure 7(b) for the case of N_CS/dV = 5 × 10¹³ cm⁻² (km s⁻¹)⁻¹ as a representative example.

In Figure 7(b), a rapid drop in ${\zeta }_{\mathrm{HCN}/\mathrm{CS}}$ (by a factor of ∼5 when ${X}_{\mathrm{HCN}}/{X}_{\mathrm{CS}}$ = 10) stands out, which damps the variation of ${\xi }_{\mathrm{HCN}/\mathrm{CS}}$. We can attribute this dive to the quite different τ between HCN(4–3) and CS(7–6); ${\tau }_{\mathrm{HCN}(4-3)}$ = 19.5–41.8, whereas ${\tau }_{\mathrm{CS}(7-6)}$ = 0.06–0.45 for the case of Figure 7(b). In this case, ${\zeta }_{\mathrm{HCN}/\mathrm{CS}}$ mostly reflects the variation of ${\tau }_{\mathrm{CS}(7-6)}$ since ${\zeta }_{\mathrm{HCN}/\mathrm{CS}}$ is now $\sim 1/{\tau }_{\mathrm{CS}(7-6)}$. Then, considering the τ of HCN(4–3) and CS(7–6) shown in Figure 4, we can deduce that the condition of "HCN(4–3) is optically thick, whereas CS(7–6) is optically thin" would be the key to realize the high ${R}_{\mathrm{HCN}/\mathrm{CS}}\;\gtrsim$ 10 observed in some AGNs. This scenario can explain the dependence of ${R}_{\mathrm{HCN}/\mathrm{CS}}$ on T_kin, as we can expect larger ${\tau }_{\mathrm{CS}(7-6)}$ (higher ${T}_{\mathrm{ex},\mathrm{CS}(7-6)}$) at higher T_kin, while ${\tau }_{\mathrm{HCN}(4-3)}$ is already substantially large regardless of any T_kin in our models. Note that ${R}_{\mathrm{HCN}/\mathrm{CS}}$ naturally converges to ∼1 as both HCN(4–3) and CS(7–6) reach the optically thick limit and are fully thermalized.

Then, we suggest that ${N}_{\mathrm{CS}}/{dV}\;\lesssim$ 5 × 10¹³ cm⁻² (km s⁻¹)⁻¹ and ${X}_{\mathrm{HCN}}/{X}_{\mathrm{CS}}\;\gtrsim$ 10 would be necessary conditions to reproduce the observed high ${R}_{\mathrm{HCN}/\mathrm{CS}}$ in AGNs. These conditions become tighter ones if AGNs have ${T}_{\mathrm{bg}}\;\gtrsim$ 20 K and ${T}_{\mathrm{kin}}\;\gtrsim$ 100 K (Figure 6). The former assumption can be justified by the high dust temperature in NGC 1068 (46 K; García-Burillo et al. 2014). As for the latter one, it has been suggested that T_kin in AGN environments is as high as several hundred kelvin even in a molecular phase (e.g., I13; Matsushita et al. 1998; Krips et al. 2008; Davies et al. 2012; Viti et al. 2014); thus, it is highly likely to be satisfied. The required ${X}_{\mathrm{HCN}}/{X}_{\mathrm{CS}}$ for AGNs (≳10) is again significantly higher than that for SB galaxies (∼3). Hence, the estimated ${\mathrm{BF}}_{\mathrm{HCN}/\mathrm{CS}}$ is at least ∼3. This ${\mathrm{BF}}_{\mathrm{HCN}/\mathrm{CS}}$ can increase to ∼10 (i.e., X_HCN/X_CS ∼ 30 in AGNs) to reproduce the high-end values of ${R}_{\mathrm{HCN}/\mathrm{CS}}\;\gtrsim$ 12 (Figure 1) under the conditions of moderately high dust temperature of ≳10 K and high T_kin such as 200 K. These boosting factors are roughly consistent with ${\mathrm{BF}}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$.

Recently, Viti et al. (2014) reported that ${n}_{{{\rm{H}}}_{2}}$ would vary even inside a single CND around an AGN. Therefore, the impact of different densities on the line ratios of our interest should be investigated as a subsequent analysis to Sections 4.2–4.3. To this end, we conducted the same analysis as shown in the previous parts but for the case of ${n}_{{{\rm{H}}}_{2}}$ = 5 × 10⁶ cm⁻³. This density roughly corresponds to the highest one predicted by Viti et al. (2014) in the CND of NGC 1068. The resultant (model-predicted) line ratios and optical depths are presented in Figures 8 and 9, respectively.

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From that figure, we found that both ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ and ${R}_{\mathrm{HCN}/\mathrm{CS}}$ tend to show higher values than those in Figure 6 when N_mol/${dV}\;\leqslant$ 5 × 10¹³ cm⁻² (km s⁻¹)⁻¹ (mol = HCO⁺ and CS), i.e., cases when emission lines are optically thin or moderately thick. Especially, ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ in Figure 8(a) exceeds 2.5 at ${T}_{\mathrm{kin}}\;\geqslant$ 100 K and X_HCN/${X}_{{\mathrm{HCO}}^{+}}$ = 10, which is comparable to the observed high-end ratios in Table 1. The higher values than the former cases with ${n}_{{{\rm{H}}}_{2}}$ = 10⁵ cm⁻³ directly reflect the highest n_cr of HCN(4–3) among the lines used here. However, when N_mol/dV is increased, one can also see that the line ratios converge to unity more quickly than for the cases in Figure 6. Enhanced excitation due to the higher ${n}_{{{\rm{H}}}_{2}}$ makes each emission line thermalized and optically thick, leading to this convergence. This is also manifested in the trend that both line ratios are less dependent on T_bg, but more sensitive to T_kin than the cases in Figure 6, which is especially prominent in ${R}_{\mathrm{HCN}/\mathrm{CS}}$ (this ratio is sensitive to the optical depth; Section 4.3). Therefore, although it depends on the line opacity, higher ${n}_{{{\rm{H}}}_{2}}$ does not necessarily correspond to higher ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ or ${R}_{\mathrm{HCN}/\mathrm{CS}}$.

Then, let us speculate more in detail about three cases of gas densities in the following: (i) ${n}_{\mathrm{AGN}}={n}_{\mathrm{SB}}$, (ii) ${n}_{\mathrm{AGN}}\lt {n}_{\mathrm{SB}}$, and (iii) n_AGN > n_SB. Here n_AGN and n_SB denote representative gas densities in AGN and SB environments, respectively. We focus on two cases of ${n}_{{{\rm{H}}}_{2}}$ as shown in Figures 6 (10⁵ cm⁻³) and 8 (5 × 10⁶ cm⁻³), and we also use only ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ during this speculation, for simplicity. Then, the higher density between n_AGN and n_SB corresponds to 5 × 10⁶ cm⁻³, whereas the lower one is 10⁵ cm⁻³ hereafter. Following the same manner, an essentially similar argument for abundance ratios can be achieved for the case of X_HCN/X_CS as well. Note that, however, we can only argue for the rough trend of relative difference in the abundance ratio between AGNs and SB galaxies in the following, because of this simple treatment. More comprehensive and quantitative comparison requires high-resolution, multi-line, and multi-species analysis to restrict the gas excitation. Such an analysis should be conducted with future observations.

(i) n_AGN = n_SB: in the case of ${n}_{{{\rm{H}}}_{2}}$ = 10⁵ cm⁻³ (Figure 6), X_HCN/${X}_{{\mathrm{HCO}}^{+}}$ is ∼a few for SB galaxies (Section 4.3). Then, we request that the boosting factor ${\mathrm{BF}}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ be at least ∼3 to explain the observed ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ in AGNs. Regarding a high-end value such as ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}\;\gtrsim$ 3 (e.g., NGC 1068 (W-knot)), ${\mathrm{BF}}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ even reaches ≳10 (i.e., X_HCN/${X}_{{\mathrm{HCO}}^{+}}\;\gtrsim$ 30). When ${n}_{{{\rm{H}}}_{2}}$ is increased to 5 × 10⁶ cm⁻³ (Figure 8), X_HCN/${X}_{{\mathrm{HCO}}^{+}}$ = 1 is sufficient to reproduce the ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ in SB galaxies (Figure 8). On the other hand, we still need ${\mathrm{BF}}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ to be a few to even ∼10 (depending on the line opacity) in order to account for the ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ in AGNs, especially the high-end values. In either case, we require the enhanced HCN abundance in AGNs to reproduce the observations.

(ii) n_AGN < n_SB: as shown above, X_HCN/${X}_{{\mathrm{HCO}}^{+}}\;\sim$ 1 is sufficient to reproduce the observed ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ in SB galaxies when ${n}_{{{\rm{H}}}_{2}}$ = 5 × 10⁶ cm⁻³ (Figure 8). However, we need to enhance the X_HCN/${X}_{{\mathrm{HCO}}^{+}}$ up to ∼10 (≳30 for the case of the high-end value) to explain the ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ in AGNs when ${n}_{{{\rm{H}}}_{2}}$ = 10⁵ cm⁻³. Hence, again it indicates substantially high X_HCN/${X}_{{\mathrm{HCO}}^{+}}$ in AGNs with ${\mathrm{BF}}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ as high as ≳10–30.

(iii) n_AGN > n_SB: in the case of ${n}_{{{\rm{H}}}_{2}}$ = 10⁵ cm⁻³, X_HCN/${X}_{{\mathrm{HCO}}^{+}}\;\sim$ 3 is sufficient to reproduce the observed ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ in SB galaxies (Figure 6). For AGNs having higher gas density of ${n}_{{{\rm{H}}}_{2}}$ = 5 × 10⁶ cm⁻³ (Figure 8), we have to restrict ourselves to ${N}_{{\mathrm{HCO}}^{+}}$/dV ≲ 5 × 10¹³ cm⁻² (km s⁻¹)⁻¹ at first because ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ converges to unity for higher column density (or τ) and thus never reaches ≳2. Under these conditions, we found that X_HCN/${X}_{{\mathrm{HCO}}^{+}}\;\sim$ 3–5 can indeed yield a high ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ of ∼1.5–2.5 especially when ${N}_{{\mathrm{HCO}}^{+}}$/dV ∼ 5 × 10¹² cm⁻² (km s⁻¹)⁻¹ and T_kin ≳ 100 K (Figure 8(a)). In this case, ${\mathrm{BF}}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ is only ∼1–1.5, i.e., no abundance variation between AGNs and SB galaxies. Note that, even with this high ${n}_{{{\rm{H}}}_{2}}$, it is still a bit challenging to reproduce the observed highest ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ of ≳3. To do so, we might need to require ${\mathrm{BF}}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ to be ∼a few even with the high ${n}_{{{\rm{H}}}_{2}}$ in AGNs. On the other hand, when ${N}_{{\mathrm{HCO}}^{+}}$/dV ∼ 5 × 10¹³ cm⁻²(km s⁻¹)⁻¹, we cannot avoid increasing ${\mathrm{BF}}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}\;\gtrsim$ a few to yield the high-end ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ in AGNs (Figure 8(b)).

As a summary of our non-LTE modelings with both ${n}_{{{\rm{H}}}_{2}}$=10⁵ cm⁻³ and 5 × 10⁶ cm⁻³, we suggest that ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$ would be enhanced in AGNs as compared to SB galaxies by several times to ≳10 times when ${n}_{{{\rm{H}}}_{2}}$ in AGNs are comparable to or lower than those in SB galaxies in order to reproduce their observed ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$. Another plausible origin might be the systematically higher gas density in AGNs than in SB galaxies, which should be coupled to the low opacity of the emission lines from AGNs, to yield the high ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$. In this case, an almost comparable ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$ is expected between AGNs and SB galaxies. However, the feasibility of such a systematic difference in gas density is unclear at this moment. Although Viti et al. (2014) suggested such differences between the kiloparsec-scale SB ring and the 100 pc scale CND of NGC 1068, what we are treating here is the rather smaller, circumnuclear-scale (likely to be ≲100 pc scale) components located at the centers of both AGN and SB galaxies; recall that we made Figure 1 only based on the high-resolution sample in Table 1. Hence, the contrast in density between AGN and circumnuclear SB environments would be smaller than in the case between AGN and kiloparsec-scale SB environments. Moreover, ∼30 pc scale measurements of HCN(4–3) and HCN(1–0) revealed that the HCN(4–3)/HCN(1–0) integrated intensity ratio is ∼0.7–1.0 in the central region of NGC 253 (the brightness temperature scale; Sakamoto et al. 2011; Meier et al. 2015), which is comparable to the values obtained within the CND of NGC 1068 (Viti et al. 2014). Knudsen et al. (2007) also proposed that the line excitation is independent of galaxy types, although this argument was based on single-dish measurements of the HCN spectral energy distribution. These results suggest that the degree of line excitation at the nuclear region of a galaxy would not depend on the specific type of that galaxy. Likely higher T_kin in AGNs would lead to lower ${n}_{{{\rm{H}}}_{2}}$ than SB galaxies if they share the same line excitation. Therefore, we hereafter prefer the scenario of the variation of underlying molecular abundances between AGNs and SB galaxies as the prime cause of the HCN enhancement. However, we should mention that we cannot discard the possibility of systematically higher ${n}_{{{\rm{H}}}_{2}}$ in AGNs than in SB galaxies, nor the possibility that the CND of NGC 1068 has specifically and remarkably high ${n}_{{{\rm{H}}}_{2}}$ as compared to other AGNs and SB galaxies, because of the simplified analysis and discussion in this work.

In this section, we briefly compare our model calculations with the observed ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ and/or ${R}_{\mathrm{HCN}/\mathrm{CS}}$ of two instructive AGNs and one buried AGN, namely, NGC 1097, NGC 1068, and NGC 4418. The characteristics of these galaxies are presented in Appendix A. Models with ${n}_{{{\rm{H}}}_{2}}$ = 10⁵ cm⁻³ (Figure 6) are employed here, but those with ${n}_{{{\rm{H}}}_{2}}$ = 5 × 10⁶ cm⁻³ yield essentially the same (qualitative) arguments. As these models were not constructed to mimic the environment of a specific galaxy, our immediate objective here is to roughly check the consistency of the trend between the model-predicted line ratios and the actual observed values.

For NGC 1097 (AGN), Hsieh et al. (2008) estimated ${N}_{{{\rm{H}}}_{2}}$ = 5.9 × 10²² cm⁻² from CO interferometric observations. Assuming X_CS = 2.5 × 10⁻⁹ (the value of the OMC-1 Extended Ridge, where ${N}_{{{\rm{H}}}_{2}}$ = 3 × 10²³ cm⁻²; Blake et al. 1987) and a line width of 250 km s⁻¹ (FWZI of HCN(4–3); I13), ${N}_{\mathrm{CS}}/{dV}$ is estimated to be ∼6 × 10¹¹ cm⁻² (km s⁻¹)⁻¹. Then, judging from Figure 4, we can expect that CS(7–6) emission is totally optically thin (${\tau }_{\mathrm{CS}(7-6)}\;\ll$ 0.1). This is consistent with the nondetection of CS(7–6) emission in our previous observations with ALMA (I13). Even if we increase X_CS by 10 times, the situation is the same. On the other hand, non-LTE analysis by I13 suggested that ${N}_{\mathrm{HCN}}/{dV}$ in NGC 1097 is ∼a few × 10¹³ cm⁻² (km s⁻¹)⁻¹; thus, HCN(4–3) would be moderately optically thick (Figure 4). The resultant X_HCN in this case is ∼10⁻⁷, which is close to the value observed in Galactic hot cores and is significantly higher than that observed in, e.g., the OMC-1 Extended Ridge (5.0 × 10⁻⁹; Blake et al. 1987). Therefore, the situation is similar to panel (e) of Figure 6 with a high ${X}_{\mathrm{HCN}}/{X}_{\mathrm{CS}}$ of ≳10. Indeed, our model using the above values shows ${R}_{\mathrm{HCN}/\mathrm{CS}}\;\gtrsim$ 20, which matches the observed value (${R}_{\mathrm{HCN}/\mathrm{CS}}\;\gt$ 12.7). Furthermore, an optical depth of HCN(1–0) estimated from the above number is ${\tau }_{\mathrm{HCN}(1-0)}\;\gtrsim$ 0.1 (T_kin = 100 K is assumed), which is consistent with the estimation by Martín et al. (2015).

However, our estimated abundance ratios such as X_HCN/${X}_{{\mathrm{HCO}}^{+}}\;\sim$ 10 are significantly higher than those obtained by Martín et al. (2015) through LTE analysis, which are ∼3. As long as we adopt a gas density that is not high enough to thermalize all of the HCN(4–3), HCO⁺(4–3), and CS(7–6) emission lines (e.g., ${n}_{{{\rm{H}}}_{2}}$ = 10⁵ cm⁻³ in Figure 6), we should require highly enhanced abundance ratios to overcome the inefficient excitation of the HCN(4–3) line (this line has the highest n_cr among the target lines; Table 2) and yield the high line ratios observed. In the case of higher ${n}_{{{\rm{H}}}_{2}}$ such as 5 × 10⁶ cm⁻³, the required abundance ratio can be as low as ∼3 for optically thin emission lines (Figures 8(a), (e)). This ratio is consistent with the measured values by Martín et al. (2015), because now the conditions of (i) LTE and (ii) optically thin emission are mostly satisfied. For optically thicker cases, we will again see a discrepancy between our model predictions and those by Martín et al. (2015). Therefore, whether our model predictions are consistent or not with the previous LTE results strongly depends on the assumed ${n}_{{{\rm{H}}}_{2}}$ and line optical depth.

In the case of NGC 1068, both the ratios of ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ and ${R}_{\mathrm{HCN}/\mathrm{CS}}$ at the precise AGN position are lower than those of NGC 1097 (AGN). We suppose that this is likely due to a larger N_mol than that of NGC 1097 (AGN), i.e., the lines are optically thicker than those of NGC 1097 (AGN). Actually, the ${N}_{{{\rm{H}}}_{2}}$ toward this AGN can be ∼6 × 10²³ cm⁻² by using the results of the CO multi-line non-LTE analysis by Viti et al. (2014) (see also the footnote in Section 2 for the estimation), which is ∼10 times larger than that of NGC 1097 (AGN). In this case, again by applying X_CS = 2.5 × 10⁻⁹ (Blake et al. 1987) and dV ∼ 200 km s⁻¹ (García-Burillo et al. 2014), we estimate N_CS/dV ∼ 1 × 10¹³ cm⁻² (km s⁻¹)⁻¹. For this N_CS/dV, CS(7–6) emission is still moderately optically thin (τ_CS ∼ 0.1; Figure 4), but is significantly optically thicker than the case of NGC 1097 (AGN). The same speculation can also be applied to HCN(4–3) and HCO⁺(4–3) emission lines. As shown in Figure 6, it is natural for line ratios to converge to unity when the paired lines get optically thicker. Moreover, since ${R}_{\mathrm{HCN}/\mathrm{CS}}$ is sensitive to T_bg, we can expect a lower value of this ratio in NGC 1068 (AGN) than in NGC 1097 (AGN) even when they share the same X_HCN/X_CS considering the much higher AGN luminosity of the former than the latter (Marinucci et al. 2012; Liu et al. 2014). Therefore, despite the rather lower ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ and ${R}_{\mathrm{HCN}/\mathrm{CS}}$ than in NGC 1097 (AGN), we still expect that the underlying ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$ and ${X}_{\mathrm{HCN}}/{X}_{\mathrm{CS}}$ are high (∼10) in NGC 1068 (AGN). Note that the line optical depths of HCN(4–3) are 40–60 even when N_HCN/dV = 1 × 10¹⁵ cm⁻² (km s⁻¹)⁻¹, which are still smaller than that observed in Arp 220W (a system of self-absorption; Scoville et al. 2015).

As for the knots of the CND of NGC 1068, panels (a), (b), (e), and (f) of Figure 6 would be good approximations for their situations if we adopt the estimated column densities by Viti et al. (2014). In those cases with T_kin > 100 K, we need the abundance ratios to be ≳30 when ${n}_{{{\rm{H}}}_{2}}$ = 10⁵ cm⁻³. As stated before, we cannot discard the possibility of no abundance enhancement in the case of panels (a) and (e) in Figure 8. However, if N_mol/dV ≳ 5 × 10¹³ cm⁻² (km s⁻¹)⁻¹ is true, we still need to enhance the abundance ratios to be ≳30 even with ${n}_{{{\rm{H}}}_{2}}$ = 5 × 10⁶ cm⁻³ to reproduce the high-end values observed in these knots (e.g., ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}\;\gtrsim$ 3). Note that we can find consistent abundance ratios with those suggested above in previous works of multi-line, multi-species non-LTE analysis (e.g., Table 8 of Viti et al. 2014). On the other hand, there seems to be an inconsistency with the predicted abundance ratios from LTE analysis (Table 5 of Viti et al. 2014), which are ∼5 for both X_HCN/${X}_{{\mathrm{HCO}}^{+}}$ and X_HCN/X_CS. For ${n}_{{{\rm{H}}}_{2}}$ of 10⁵ cm⁻³ (Figure 6), again we should enhance X_HCN to overcome the inefficient excitation of HCN(4–3). For the higher ${n}_{{{\rm{H}}}_{2}}$ of 5 × 10⁶ cm⁻³ (Figure 8), our results are consistent with those from the LTE analysis as long as emission lines are optically thin or moderately thick (see also Figure 9), but the discrepancy becomes prominent for optically thicker cases because now the condition of optically thin emission assumed in the LTE analysis (Viti et al. 2014) breaks. In such cases, we eventually require enhanced X_HCN/${X}_{{\mathrm{HCO}}^{+}}$ of, e.g., <30, as already mentioned, e.g., X_HCN/${X}_{{\mathrm{HCO}}^{+}}$ of ≳ 30 as already addressed.

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In the case of NGC 4418, the same argument of the optical depth effect as for NGC 1068 (AGN) holds with a stronger basis, because this galaxy hosts a heavily obscured compact nucleus (${N}_{{{\rm{H}}}_{2}}\;\gtrsim$ 10²⁶ cm⁻²; Sakamoto et al. 2013). The distinctively lower ${R}_{\mathrm{HCN}/\mathrm{CS}}$ than normal AGNs favors this view as this ratio is highly sensitive to the optical depth of the CS(7–6) emission line (Figure 7). Regarding ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$, we mention that this ratio is slightly increased to ∼2 when observed at 05–80'' pc resolution (Sakamoto et al. 2013), which is a comparable value to other AGNs. However, whether an enhanced X_HCN/${X}_{{\mathrm{HCO}}^{+}}$ such as 10 is really necessary to account for this relatively high ratio is unclear, because this galaxy exhibits a prominent v = 1 state HCN emission line (IR pumping; Sakamoto et al. 2010). Radiative transfer modelings involving the IR pumping are thus inevitable to elucidate the validity of our diagram in such a deeply embedded nucleus that the IR pumping must affect the ratio.

As the last of Section 4, we here predict HCN-to-HCO⁺ and HCN-to-CS integrated intensity ratios at other transitions based on our RADEX modelings, by varying ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$ and ${X}_{\mathrm{HCN}}/{X}_{\mathrm{CS}}$. Illustrative cases are shown in Figure 10 as a function of the upper rotational state (J_u) of HCN. As for CS, we coupled a transition with the closest frequency to each HCN transition (e.g., HCN(1–0) and CS(2–1), HCN(4–3) and CS(7–6)). We hereafter call this type of diagram a spectral ratio distribution (SRD). Note that we do not intend to predict the line ratio of a specific galaxy in Figure 10, but to understand a qualitative feature of these ratios at various transitions.

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By inspecting the SRD, one can see that enhanced ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$ (or ${X}_{\mathrm{HCN}}/{X}_{\mathrm{CS}}$) correspondingly produces higher line ratios at any transition. Therefore, if X_HCN/${X}_{{\mathrm{HCO}}^{+}}$ and X_HCN/X_CS are truly enhanced more in AGNs than in SB galaxies, and if they share the comparable excitation (e.g., ${n}_{{{\rm{H}}}_{2}}$ and T_kin), we predict that AGNs would show higher line ratios not only at a single transition but also at other transitions than SB galaxies. Indeed, NGC 1097 and NGC 1068 show higher HCN(1–0)/HCO⁺(1–0) and HCN(1–0)/CS(2–1) ratios than SB galaxies (e.g., Kohno et al. 2008; Nakajima et al. 2011; Martín et al. 2015).

From this perspective, remarkable inconsistency can stem from NGC 7469, which shows an HCN(1–0)/HCO⁺(1–0) ratio of ∼0.6 when observed with the Nobeyama Millimeter Array (∼6'' = 2 kpc aperture; Kohno 2005), whereas it shows HCN(4–3)/HCO⁺(4–3) = 1.1 (∼150 pc aperture; Table 1). This inconsistency can be reconciled when we measure J = 1–0 lines at a higher resolution to selectively probe the CND of NGC 7469 as much as possible, because this galaxy hosts a prominent SB ring with ∼1 kpc in diameter that can be a strong source of spectral contamination; the HCN(4–3)/HCO⁺(4–3) ratio at the SB ring is as low as ∼0.5. In fact, ∼150 pc aperture measurements of J = 1–0 transitions newly obtained with ALMA reveal HCN(1–0)/HCO⁺(1–0) > 1 at the AGN position (T. Izumi et al. 2016, in preparation). This improved spatial resolution is still larger than but closer to the expected size of the XDR in NGC 7469 (∼80 pc in diameter; Izumi et al. 2015). With this in mind, we here suppose that the low HCN(1–0)/HCO⁺(1–0) ratio (<1) observed in NGC 2273 and NGC 4051 (Sani et al. 2012) would be (at least partly) due to severe spectral contamination from the surrounding SB regions. This can be the same situation as we discussed for NGC 1365 and NGC 4945 in Section 3. These low line ratios were measured in the central 3'' regions (Sani et al. 2012), which correspond to 390 pc for NGC 2273 and 150 pc for NGC 4051. Indeed, the equivalent widths of the 11.3 μm PAH emission in NGC 2273 (330 nm with ∼500 pc slit; Alonso-Herrero et al. 2014) and NGC 4051 (95 nm with 25 pc slit; Alonso-Herrero et al. 2016) are significantly larger than those of NGC 7469 or NGC 1068 (Section 3). Moreover, the 2–10 keV X-ray luminosities of NGC 2273 (log (${L}_{2-10\mathrm{keV}}$/erg s⁻¹) = 42.7; Marinucci et al. 2012) and NGC 4051 (log (${L}_{2-10\mathrm{keV}}$/erg s⁻¹) =41.1; Liu et al. 2014) are ∼3 and ∼130 times smaller than that of NGC 7469 (log (${L}_{2-10\mathrm{keV}}$/erg s⁻¹) = 43.2; Liu et al. 2014), respectively. Therefore, we can expect that the extent of the hypothesized XDRs is much smaller in NGC 2273 and NGC 4051 than that of NGC 7469 (Izumi et al. 2015) and hence than the above-mentioned 3'' areas where the line ratios were measured. On the other hand, again we cannot discard the possibility that these galaxies have less dense gas than AGNs with high line ratios (see also Section 4.4). Multi-line, multi-species modelings are indeed required to disentangle these scenarios.

It has been suggested that the enhanced HCN(1–0) intensity observed in AGNs is due to abnormal chemistry realized in XDRs (e.g., Kohno 2005; Krips et al. 2008). The molecular abundances and the resultant column-integrated line intensities of our target molecules under steady-state gas-phase XDRs and PDRs were extensively modeled by Meijerink & Spaans (2005) and Meijerink et al. (2007). In their XDR models, ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ can exceed unity only at the surface (N_H ≲ 10^22.5 cm⁻²) of low- to moderate-density gas (n_H ≲ 10⁵ cm⁻³), where a high X-ray flux (e.g., F_X ≳ 10 erg s⁻¹ cm⁻²) can be expected. However, for a larger N_H where F_X is attenuated, they predicted ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}\;\lt$ 1. This is due to the fact that the range of the ionization rate over which ${X}_{{\mathrm{HCO}}^{+}}$ is high is much wider than that of HCN under X-ray ionization chemistry (Lepp & Dalgarno 1996). Note that one weak point in this model would be that the predicted line intensities are considerably low at the region of N_H ≲ 10^22.5 cm⁻² (Meijerink et al. 2006, 2007), which seems to be inconsistent with the prominent HCN(4–3) and HCO⁺(4–3) emissions observed in AGNs.

Another possible scenario is a high-temperature chemistry, under which neutral–neutral reactions with high reaction barriers are enhanced in general (e.g., Harada et al. 2010). This kind of chemistry is efficient in, e.g., hot core-like regions and mechanically dominated regions. For example, a formation path of CN + ${{\rm{H}}}_{2}\;\to$ HCN + H has a barrier of ∼960 K (KIDA).²² This reaction can be efficient at T_kin ≳ 300 K (Harada et al. 2010). Indeed, such a high temperature can likely be expected in AGNs (e.g., Krips et al. 2008; Davies et al. 2012). At that temperature, ${X}_{{\mathrm{HCO}}^{+}}$ can be somewhat reduced owing to an activated reaction with H₂O to form H₃O⁺. What is important is that this chemistry can be complemented to conventional XDR (ionization) models (Harada et al. 2013), where we can expect much higher gas temperature owing to efficient X-ray heating than in SB environments (PDR). Interestingly, recent high-resolution VLA observations revealed a higher fractional abundance of NH₃ in the very nuclear region of NGC 3079 (a type 2 AGN; Miyamoto et al. 2015) than in SB galaxies (Takano et al. 2013, and references therein). NH₃ is also a typical molecule efficiently formed in high-temperature environments. We should note that evaporation from dust grains can also increase the ${X}_{{\mathrm{NH}}_{3}}$, especially at lower temperature (≲100 K; Rodgers & Charnley 2001), which will enhance the X_HCN via subsequent nitrogen reactions. However, at low temperature, ion-neutral reactions are so quick that we can envision, e.g., C⁺ + H₂O $\to$ HCO⁺ + H, which will efficiently enhance the ${X}_{{\mathrm{HCO}}^{+}}$ (O-bearing species such as H₂O should also be ejected from dust). This would reduce ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$, which seems to be inconsistent with the high ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$ suggested in Section 4; thus, the dust grain reactions will not be very critical.

Keeping these models in mind, we hereafter present a possible interpretation of the ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ in three instructive AGNs and two SB galaxies, namely, NGC 7469, NGC 1068, NGC 1097, NGC 253, and M82.

(i) NGC 7469—we first discuss the ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ measured at the AGN position of NGC 7469 (denoted as N7469 (AGN) in Figure 1), which is comparable to those of some SB galaxies such as NGC 253 despite its orders of magnitude higher X-ray luminosity (log ${L}_{2-10\mathrm{keV}}$ = 43.2; Liu et al. 2014). No morphological and kinematic signatures of a jet–ISM interaction have been found in the CND of NGC 7469 through the high-resolution observations of H₂ and Brγ emission lines (Hicks et al. 2009; Müller-Sánchez et al. 2011), and then mechanical heating might not play an important role in this galaxy, although Lonsdale et al. (2003) found a core jet-like structure at a radio wavelength. Regarding the origin of the relatively low ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$, Izumi et al. (2015) pointed out the compactness of the XDR. They estimated the spatial extent of the XDR and the region where F_X ≳ 10 erg s⁻¹ cm⁻² (Meijerink et al. 2007) to be ∼42 pc and ∼35 pc in radius, respectively. Therefore, not a poor but a still relatively large observing beam employed in the NGC 7469 observations (∼150 pc; Table 1) might have picked up line fluxes emanating from the extended SB region, which would have resulted in reducing the ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ of this AGN.

(ii) NGC 1068—the X-ray luminosity of this AGN (log ${L}_{2-10\mathrm{keV}}$ = 43.0; Marinucci et al. 2012) is quite comparable to that of NGC 7469. The degree of contamination from PDRs and SNe (i.e., stellar feedback) to the observed line ratio is unclear, although we expect that it is relatively low by considering the moderate stellar age inside the CND (200–300 Myr; Davies et al. 2007). This age is much older than those of SB galaxies, e.g., NGC 253 (∼6 Myr; Fernández-Ontiveros et al. 2009) and M82 (∼10 Myr; Förster Schreiber et al. 2003). We found a relatively high ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ of ∼1.5 at its precise AGN position (denoted as N1068 (AGN) in Figure 1), which is well consistent with the results in García-Burillo et al. (2014). The ratio was measured with a single synthesized beam of 35 pc. García-Burillo et al. (2014) also reported a correlation between the spatially resolved HCN(4–3)-to-CO(3–2) integrated intensity ratio ($\equiv {R}_{\mathrm{HCN}/\mathrm{CO}}$) and the 6–8 keV X-ray flux ($\equiv {R}_{{\rm{X}}-\mathrm{ray}/\mathrm{CO}}$) across the CND of NGC 1068. The same trend can also be found in HCO⁺(4–3), SiO(2–1), and CN(2–1) (García-Burillo et al. 2010, 2014), which suggests that the whole CND is a giant XDR. However, again according to Meijerink et al. (2007), to reproduce ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}\;\gt$ 1 at this AGN position is quite difficult since the ${N}_{{{\rm{H}}}_{2}}$ toward the AGN position expected from the results of the CO multi-line non-LTE analysis by Viti et al. (2014) is ∼6 × 10²³ cm⁻² (see the footnote of Section 2 for the assumptions in this estimation). To interpret the high ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ of N1068 (AGN) is thus not straightforward at all. We might have to add some other mechanisms such as high-temperature chemistry (Harada et al. 2010, 2013) to forcibly enhance ${X}_{\mathrm{HCN}}$.

As for the ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ measured at other positions, e.g., the east/west knots of the CND of NGC 1068 (namely, N1068 (E-knot) and N1068 (W-knot) in Figure 1), we found that they are 2 times higher than that of N1068 (AGN). This result is a bit surprising since the ratio increases as it recedes from the nucleus (=the X-ray peak). We speculate that the cause of this enhancement at the E/W-knots is likely related to that for the prominent 2.12 μm H₂ emission there (e.g., Müller Sánchez et al. 2009). The ${R}_{\mathrm{HCN}/\mathrm{CO}}$–${R}_{{\rm{X}}-\mathrm{ray}/\mathrm{CO}}$ correlation shown above and the very uniform distribution of the H₂ λ2.25 μm/λ2.12 μm ratio across the CND (Riffel et al. 2014) seem to support a scenario in which these emissions are primarily due to (nondirectional) X-ray heating (see also Galliano et al. 2003). Note that the H₂ line ratio at the CND (∼0.1) is a typical value for thermal processes by X-ray and/or shock heating (Mouri 1994). In this scenario, we might be observing HCN and HCO⁺ emissions that emerged from molecular clouds directly illuminated by X-ray radiation (i.e., clouds directly seen by the AGN itself, not strongly intercepted by a dusty torus) at the east/west knots, considering the geometry of the type 2 nucleus (the position angle of the almost edge-on H₂O maser disk is −45°; Greenhill et al. 1996) and the surrounding CND (the inclination angle is ∼41°; García-Burillo et al. 2014). If so, the X-ray fluxes received at the east/west knots are quite high because of little attenuation, which can result in efficient X-ray heating and thus high ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$ owing to, e.g., high-temperature neutral–neutral reactions (Harada et al. 2010, 2013). Indeed, if there is no attenuation, we can expect that F_X is as high as ∼100 erg s⁻¹ cm⁻² even at 30 pc away from the nucleus. According to the model calculation by Meijerink et al. (2007), this high X-ray flux can heat up the gas with ${n}_{{{\rm{H}}}_{2}}$ =10^5–6 cm⁻³ to $\gg$ 100 K.

On the other hand, the radial expansion of the CND of NGC 1068 (Krips et al. 2011; Barbosa et al. 2014) would support the existence of shocks and/or a jet–ISM (and possibly outflow–ISM) interaction, which can also contribute to excite the H₂ molecule. We especially mention that García-Burillo et al. (2014) suggested that as much as ∼50% of CO(3–2) emission stems from the outflowing component, which extends out to ∼400 pc away from the nucleus. Hence, there is at least a certain portion of the region dominated by mechanical heating (MDR; e.g., Meijerink et al. 2011; Kazandjian et al. 2012, 2015) in the CND. Again high gas temperature chemistry would be responsible for the high ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ in this case. Indeed, we can expect the maximum temperature of ${n}_{{{\rm{H}}}_{2}}$ = 10⁵ cm⁻³ gas to reach as high as ∼2500 K for C-shock when the shock velocity is >20 km s⁻¹ (Flower & Pineau des Forêts 2013), which is high enough to produce prominent H₂ emission. Such a high temperature is well beyond the reaction barrier of HCN formation from CN; thus, HCN can be abnormally abundant. According to the model calculation of molecular abundances in MDRs by Kazandjian et al. (2012), X_HCN/${X}_{{\mathrm{HCO}}^{+}}$ can well exceed 100 when the input mechanical energy is ≳10⁻¹⁸ erg s⁻¹ cm⁻³, which is large enough to produce high ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$. Interestingly, the ${R}_{\mathrm{HCN}/\mathrm{CO}}$–${R}_{{\rm{X}}-\mathrm{ray}/\mathrm{CO}}$ correlation in NGC 1068 has large scatter at lower ${R}_{{\rm{X}}-\mathrm{ray}/\mathrm{CO}}$, i.e., away from the AGN (García-Burillo et al. 2014), suggesting the importance of other heating mechanisms than X-ray radiation for driving the underlying chemistry. Indeed, the line velocity dispersion is high at the east/west knots (García-Burillo et al. 2014). In this scenario, it would also be probable that the outflow/shock velocity is so high at N1068 (AGN) that the shock wave provides ionized gas to increase ions (e.g., Dickinson et al. 1980; Elitzur 1983; Koo & Moon 1997; Rawlings et al. 2004), which would potentially result in a depressed ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$ near the nucleus.

(iii) NGC 1097—another remarkable example is the ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ measured at the AGN position of NGC 1097 (N1097 (AGN) in Figure 1). As we stated in Section 4.5, we would need enhanced X_HCN/${X}_{{\mathrm{HCO}}^{+}}$ compared to SB galaxies to yield this high ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ when ${n}_{{{\rm{H}}}_{2}}\;\sim$ 10⁵ cm⁻³ (I13). However, judging from the very low X-ray luminosity (log ${L}_{2-10\mathrm{keV}}$ = 40.8; Liu et al. 2014), the XDR chemistry seems not to play an important role in NGC 1097 (I13). Rather, considering the current spatial resolution (94 pc) for the observations (Table 1), there is another possibility that the line-emitting region is very close to the AGN where we can still expect a high X-ray flux, or a high energy deposition rate per particle (e.g., Maloney et al. 1996). Indeed, the line luminosity of HCN(4–3) in NGC 1097 is totally comparable to that of the individual molecular clump seen at the center of NGC 253 (∼20 pc in size; Sakamoto et al. 2011), suggesting the similarly small size of the line-emitting region in NGC 1097. Radiative feedback from SB activity would be discarded considering the high gas temperature (≳100 K) suggested for the molecular phase (I13; Beirão et al. 2012). On the other hand, mechanical heating can be another choice, which is supported by the detection of a compact radio jet (largest beam-convolved and projected size is ∼90 pc; Thean et al. 2000). A tentative gradient of the HCN(1–0)/HCO⁺(1–0) line ratio inside the CND (lower toward the center, and higher toward the outer edge of the CND; Martín et al. 2015) would also support this scenario, although Martín et al. (2015) did not fully resolve the CND. Note that an inferred SN rate is quite low as O(10⁻⁴) yr⁻¹ (Davies et al. 2007); thus, we can neglect its influence on the chemistry.

(iv) NGC 253 and M82—as for the SB galaxies, the predicted ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ in Meijerink et al. (2007) is almost consistent with, but seems to be a factor of a few higher than, the observed values. This can be explained by an enhanced cosmic-ray ionization rate due to frequent SNe, which would increase ${X}_{{\mathrm{HCO}}^{+}}$ even in a dense molecular cloud (e.g., Bayet et al. 2011; Meijerink et al. 2011; Aladro et al. 2013). If this is true, ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ can reflect the evolutionary phase of SB activity. The lower ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ in M82 than in NGC 253 follows this scenario well (I13; Krips et al. 2008), although we might be able to construct some counterarguments as well (Papadopoulos 2007; Zhang et al. 2014). We mention that mechanical heating seems to be responsible for the relatively high ${R}_{\mathrm{HCN}/{\mathrm{HCO}}^{+}}$ in young SB galaxy NGC 253 (SB age ∼6 Myr; Fernández-Ontiveros et al. 2009), whose ratio is totally comparable to those of N7469 (AGN) and N1068 (AGN). Indeed, this heating mechanism is claimed to be important in NGC 253 (Rosenberg et al. 2014a).

The chemistry of PDRs in high-metallicity (i.e., Z $\geqslant$ Z_⊙, where Z_⊙ denotes the solar metallicity) environments was modeled by, e.g., Bayet et al. (2012). They showed that X_HCN increases with metallicity, whereas ${X}_{{\mathrm{HCO}}^{+}}$ and X_CS are insensitive to the variation, when FUV radiation field and cosmic-ray ionization (mimicking X-ray ionization) rate are fixed. From the observational side, Davis et al. (2013) found a positive correlation between gas-phase metallicity and ${N}_{\mathrm{HCN}}/{N}_{\mathrm{CS}}$ in the sample of early-type galaxies, metal-rich spirals, and SB galaxies, which supported the Bayet et al. prediction. However, if we regard 12 + $\mathrm{log}$ [O/H] as an indicator of the gas-phase metallicity, NGC 1068, NGC 1097, NGC 253, and M82 show almost the same value (Galliano et al. 2008, and references therein). Thus, we suggest that the metallicity is not an important factor for the submm-HCN diagram at least when we treat already chemically evolved systems such as the central regions of galaxies studied in this work.

On the other hand, it seems likely that an overabundance of elemental nitrogen observed in AGNs (e.g., Storchi-Bergmann 1991) can contribute to the HCN enhancement, since X_HCN depends on the elemental abundance of nitrogen (Bayet et al. 2008). Interestingly, the HCN(1–0)/HCO⁺(1–0) line ratio is ∼4 times lower in the LMC than in nearby massive SB galaxies (Anderson et al. 2014). Although this can be partly reconciled by the different gas density in each galaxy, we can also expect that ∼2–3 times lower N/O elemental abundance ratio in the LMC than in the solar-metallicity Galactic environment (Hunter et al. 2009) would have influenced the ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$.

This mechanism must be vital for vibrationally excited lines of these dense gas tracers such as HCN(v₂ = 1^1f, J = 4–3), since their energy levels (∼1000 K) are too high to be collisionally excited. Although the limited extragalactic detections of the vibrationally excited lines (Sakamoto et al. 2010; Imanishi & Nakanishi 2013a; Aalto et al. 2015b) limit our discussion on this topic, it would at least influence the rotational population and thus the line intensity of HCN in NGC 4418 (Sakamoto et al. 2010). Meanwhile, HCN(v₂ = 1^1f, J = 4–3) emission was not detected in NGC 7469 (Izumi et al. 2015) in spite of its high IR luminosity of ${L}_{8-1000\mu {\rm{m}}}$ = 10^11.4 L_⊙, which is comparable to that of NGC 4418 (Sanders et al. 2003). This would be due to the optical depth effect; we cannot detect a fully optically thin emission even if it surely exists. The column density of HCN(v₂ = 1^1f, J = 4–3) would not be so large along the line of sight toward the CND of NGC 7469 that the emission is optically thin. We speculate that the vibrationally excited lines can only be visible in compact (i.e., close to a warm IR source), heavily obscured (i.e., high column density) nuclei such as in NGC 4418, where we can expect that HCN(v₂ = 1^1f, J = 4–3) line emission is (moderately) optically thick.

This can be important since ${X}_{\mathrm{HCN}}/{X}_{{\mathrm{HCO}}^{+}}$ is predicted to be highly time dependent with possible variations of orders of magnitude, especially when $\geqslant$10⁴ yr has passed from the ignition of chemical reactions (Meijerink et al. 2013). This mechanism would influence the chemistry more in nuclear regions of galaxies than in quiescent regions, since the former region is more dynamic than the latter. The time evolution is typically studied via time clock molecules such as sulfur-bearing species. Indeed, while ${N}_{\mathrm{CS}}/{N}_{\mathrm{SO}}$ in the CNDs of NGC 1097 and NGC 1068 estimated in LTE analysis are ∼2–3 (Martín et al. 2015; Nakajima et al. 2015), it is just ∼0.2 in the Orion hot core (Table 2 in Esplugues et al. 2014). If we regard the above CNDs as influenced by mechanical heating, and the abundances of these species in NGC 1097 and NGC 1068 depend on the evolution of hot core-like systems, this result indicates that NGC 1097 and NGC 1068 are more evolved systems than the Orion hot core (e.g., Charnley 1997; Esplugues et al. 2014). Future sensitive and systematic study of time clock species will help us understand the chemical evolution in AGNs.

Undoubtedly this is one of the best-studied Seyfert galaxies hosting a luminous type 2 AGN (${L}_{2-10\mathrm{keV}}$ = 10^43.0 erg s⁻¹; Marinucci et al. 2012). Broad Balmer emission lines are observed in the polarized light (Antonucci & Miller 1985), which is a supportive evidence for the unified model of AGNs (Antonucci 1993). In X-rays, this AGN is known to be substantially Compton thick with N_H ∼ 10²⁵ cm⁻² along the line of sight (e.g., Matt et al. 2000; Marinucci et al. 2012, 2016). NGC 1068 also hosts a ring- or arm-like SB region with a diameter of ∼30'' (Telesco & Decher 1988). Several unbiased line surveys with single-dish telescopes and/or interferometric observations of individual molecular line emissions at millimeter/submillimeter have been conducted toward this AGN (e.g., Usero et al. 2004; Krips et al. 2011; Aladro et al. 2013; García-Burillo et al. 2014; Takano et al. 2014; Nakajima et al. 2015). The CND is bright not only in CO but also, and more distinctively, in dense gas tracers such as HCN and HCO⁺ lines (e.g., Schinnerer et al. 2000; Kohno et al. 2008; García-Burillo et al. 2014). This CND mainly consists of two bright knots, namely, east and west knots, which are ∼100 pc away from the exact location of the AGN. Interestingly, the HCN(4–3)/HCO⁺(4–3) line ratio is two times higher at these knots than at the AGN location (García-Burillo et al. 2014). NGC 1068 is also known to possess a radio jet, ionized outflow, and prominent molecular outflow, which might be interacting with the CND (e.g., Krips et al. 2011).

This galaxy hosts a low-luminosity type 1 AGN (LLAGN, ${L}_{2-10\mathrm{keV}}$ = 10^40.8 erg s⁻¹; Liu et al. 2014) as evidenced by double-peaked broad Balmer emission lines with time variability (FWHM ∼ 7500 km s⁻¹; Storchi-Bergmann et al. 1997). Interestingly, NGC 1097 also showed a transient phenomenon from a LINER nucleus to a type 1 AGN. This LLAGN and the CND are surrounded by a circumnuclear SB ring with a radius of ∼10'' = 700 pc (Barth et al. 1995). At infrared, Mason et al. (2007) reported the absence of a 3.3 μm PAH feature at sub-arcsecond scale in the CND, which is typical for a strong radiation field. But attempts to fit the IR data with clumpy torus models failed, which led Mason et al. (2007) to conclude that the torus is absent or weak. Alternatively, they argued that a nuclear star-forming cluster, which is likely coexisting with the LLAGN (Storchi-Bergmann et al. 2005), would be dominant at least in mid-IR (MIR) energetics. Both the CND and the SB ring accompany a large amount of molecular gas prominent in, e.g., CO, HCN, and HCO⁺ emissions (e.g., I13; Hsieh et al. 2008, 2012; Kohno et al. 2003). It has been claimed that the CND is warm even in the molecular phase (∼several hundred kelvin; I13; Beirão et al. 2012), which would be heated by the AGN radiatively and/or mechanically.

This archetypal barred spiral galaxy hosts both a Seyfert 1.5 nucleus (${L}_{2-10\mathrm{keV}}$ = 10^42.3 erg s⁻¹; Liu et al. 2014) with time variability (Walton et al. 2014) and a circumnuclear SB ring with a radius of 5''–10''. Several star clusters are embedded inside the SB ring (Galliano et al. 2005). A comprehensive review of this galaxy can be found in Lindblad (1999). Alonso-Herrero et al. (2012b) estimated that the AGN contributes only ∼5% of the total IR emission in the central ∼5 kpc, although it seems to dominate the energetics at λ ≲ 24 μm. Previous CO multi-transition observations revealed that most of their emission comes from the SB ring, and little is from around the Seyfert nucleus (Sakamoto et al. 2007). This suggests that the primary heating source of molecular gas is the SB activity in the ring, rather than the AGN, although the spatial distribution of dense gas tracers such as HCN and HCO⁺ is currently unknown. High-resolution observations of them are thus desirable.

This galaxy contains an AGN as evidenced by its strong and rapidly variable X-ray emission (Guainazzi et al. 2000; Puccetti et al. 2014). The intrinsic 2–10 keV luminosity is estimated to be ${L}_{2-10\mathrm{keV}}$ = 10^42.5 erg s⁻¹ (Guainazzi et al. 2000). The AGN is classified as Seyfert 2 with a huge amount of obscuring material (N_H = 10^24.5 cm⁻²; Puccetti et al. 2014). It has also been found that NGC 4945 exhibits a prominent SB activity including a large IR luminosity and IRAS color similar to a giant H ii region, but almost an edge-on view (i = 78°) has prevented detailed study at optical wavelengths. Marconi et al. (2000) reported that the spatial distribution of the starbursting region is ring-like with a radius of ∼25 (50 pc), which is visible most clearly in a Paα map. Chou et al. (2007) argued that this SB activity, rather than the AGN, is the dominant heating source for dust in the nuclear region. Furthermore, interferometric HCN(1–0), HCO⁺(1–0), and HNC(1–0) observations revealed that these dense gas tracers are emitted from an inclined rotating disk-like component with a radius of ∼4'' (Cunningham & Whiteoak 2005), i.e., emanating from both around the AGN and the SB ring. Higher-resolution observations are thus necessary to obtain a reliable spectrum of each component.

The existence of a type 1 AGN (${L}_{2-10\mathrm{keV}}$ = 10^43.2 erg s⁻¹; Liu et al. 2014) in this galaxy is evidenced by broad Balmer emission lines (FWHM ∼ 4370 km s⁻¹; e.g., Peterson et al. 2014) with time variability (Bonatto & Pastoriza 1990; Collier et al. 1998). Time variability is also confirmed by UV and X-ray observations (e.g., Nandra et al. 2000; Scott et al. 2005). A core jet-like structure (Lonsdale et al. 2003) and ionized outflows (Blustin et al. 2007) have been found, although high-resolution H₂ observations could not detect any signature of a jet–ISM (or outflow–ISM) interaction in the CND (e.g., Hicks et al. 2009). Similar to some other Seyfert galaxies, the AGN and the CND of NGC 7469 are surrounded by a prominent SB ring with a radius of 15–25 (Soifer et al. 2003; Díaz-Santos et al. 2007), which accounts for two-thirds of the bolometric luminosity of the galaxy (Genzel et al. 1995). Inside the nuclear r ≲ 130 pc region, Davies et al. (2007) reported the existence of star clusters with modest ages of 110–190 Myr. As for the molecular phase, Izumi et al. (2015) reported the significant concentration of HCN(4–3), HCO⁺(4–3), and CS(7–6) line emissions toward the CND.

This is a luminous infrared galaxy (LIRG) with an infrared luminosity of ${L}_{8-1000\mu {\rm{m}}}\;\sim$ 10¹¹ L_⊙ (Imanishi et al. 2004). NGC 4418 shows no clear AGN signature at optical and X-ray (Armus et al. 1989; Maiolino et al. 2003), However, the 5–23 μm MIR spectrum shows a typical feature of an obscured AGN (Spoon et al. 2001). Furthermore, Imanishi et al. (2004) found that the nuclear star-forming activity estimated from PAH emission can account for only 1/50 of the total IR luminosity, suggesting that the primary energy source is deeply embedded in dust and gas. Indeed, Sakamoto et al. (2013) estimated that the dust continuum opacity at 860 μm is surprisingly as large as ${\tau }_{860\mu {\rm{m}}}\;\sim$ 1, suggesting a concordantly large hydrogen column density of N_H ∼ 10²⁶ cm⁻². As for the hidden energy source, Imanishi et al. (2004) also suggested that the observed large equivalent width of H₂ emission and a high HCN(1–0)/HCO⁺(1–0) ratio at the nucleus can be well explained by the presence of a strong X-ray source such as an AGN. Note that shock heating due to in/outflows detected by molecular observations (e.g., González-Alfonso et al. 2012; Sakamoto et al. 2013) would also influence these features. Considering the above, we classify this galaxy as a buried AGN in this paper, although the evidence is still tentative.

This is a ULIRG with ${L}_{8-1000\mu {\rm{m}}}$ = 10^12.1 L_⊙ (Imanishi et al. 2006), consisting of two nuclei with a separation of ∼10'' (Kim et al. 2002). Optical spectroscopy classified this object as a non-Seyfert (LINER or H ii region; Veilleux et al. 1999), but the nuclear source is obscured as indicated by a large silicate optical depth of ∼2.5–3.0 (Imanishi et al. 2007a). IR observations, on the other hand, suggest that the energy source of the northeast nucleus is very compact, which is further characterized by weak PAH emissions and a steep temperature gradient (Imanishi et al. 2006, 2007a). These are characteristic features to buried AGNs; thus, we classify this object as such. Note that IRAS 12127-1412 has a 1.4 GHz radio luminosity that is ∼4 times brighter than the value expected for a star-forming galaxy (Spoon & Holt 2009), suggesting the existence of an AGN jet. This hypothesized jet may explain the observed outflow signatures in the [Ne iii] line (Spoon & Holt 2009) as well.

This galaxy, as well as NGC 253, is one of the most studied galaxies after the Milky Way. It is famous for its intense star formation (SFR ∼ 10 M_⊙ yr⁻¹; Förster Schreiber et al. 2003) and a prominent superwind at many wavelengths (e.g., Bregman et al. 1995; Ohyama et al. 2002). The current SB region extends over ∼500 pc centered on the nucleus (SB core), which consists of four high surface brightness clumps denoted as A, C, D, and E in O'Connell & Mangano (1978). The bolometric luminosity of the SB core is ∼10^10.8 L_⊙, half of which is emanating from OB stars (Förster Schreiber et al. 2003). Higher-resolution Hubble Space Telescope observations revealed that ∼200 super star clusters exist in that nuclear region (e.g., Melo et al. 2005). The SB history was extensively modeled by Rieke et al. (1993) and Förster Schreiber et al. (2003), who predicted that the bursts occurred ∼10 and 5 Myr ago, each lasting for a few Myr. The estimated age is consistent with the observed molecular properties, suggesting that M82 hosts an evolved SB whose energetics is dominated by radiative processes (i.e., PDRs) rather than shocks (e.g., Aladro et al. 2011). Note that it has been postulated that M82 hosts an intermediate-mass black hole located ∼600 pc away from the nucleus (e.g., Patruno et al. 2006; Pasham et al. 2014).

The star-forming region of this nearest and IR-brightest SB galaxy is located in the central r ≲ 500 pc (e.g., Scoville et al. 1985). High-resolution millimeter/submillimeter observations revealed that molecular gas in the nuclear region is distributed along a 700 × 200 pc bar-like structure containing several dense clumps (e.g., Paglione et al. 1995; Sakamoto et al. 2006a, 2011). The SB activity has been taking place for at least a few Myr, producing an IR luminosity of ∼10^10.5 L_⊙ (Telesco & Harper 1980), or an SFR of ∼5 M_⊙ yr⁻¹ assuming a Kennicutt–Schmidt relation (Kennicutt 1998). The nuclear region consists of a number of massive stellar clusters (e.g., Fernández-Ontiveros et al. 2009), some of which have a radio counterpart (Ulvestad & Antonucci 1997). Assuming an instantaneous SB, the age of the stellar clusters is estimated to be ∼6 Myr, whereas they would consist of young (≲4 Myr) and old (∼10 Myr) populations in actuality (Fernández-Ontiveros et al. 2009). Prominent ionized and molecular outflows are emerging from this region as well (e.g., Weaver et al. 2002; Bolatto et al. 2013). Considering the relatively young stellar age, NGC 253 would be in an early phase of an SB evolution. Indeed, Martín et al. (2006) suggested that the low-velocity shocks rather than UV radiation would dominate the molecular chemistry in NGC 253.

The merging system NGC 1614 is classified as a LIRG (${L}_{8-1000\mu {\rm{m}}}$ = 10^11.6L_⊙; Imanishi & Nakanishi 2013b), which has a bright optical center and two spiral arms at scales of a few kiloparsecs. At optical wavelengths, it is classified as an SB galaxy (Veilleux et al. 1995; Véron-Cetty & Véron 2010). Indeed, a 2.5–30 μm IR spectrum, prominent 3.3 μm PAH emission and 4.05 μm Brα emission (and their ratios to IR luminosity), and a 2.3 μm stellar CO absorption feature are typical of an SB galaxy (Alonso-Herrero et al. 2001; Brandl et al. 2006; Imanishi et al. 2010), and there is no clear indication of AGN activity. The star-forming activity is significantly concentrated toward the nuclear region as indicated by an SB ring with a radius of ∼300 pc clearly visible at Paα emission (Alonso-Herrero et al. 2001) and radio emission (e.g., Olsson et al. 2010). A low HCN(1–0)/HCO⁺(1–0) ratio is also supportive of dominance of SB activity (Costagliola et al. 2011).

The late-stage merging system NGC 3256 is the most luminous galaxy within 40 Mpc with ${L}_{8-1000\mu {\rm{m}}}$ = 10^11.6 L_⊙ (Sanders et al. 2003), which has a double nucleus with a separation of ∼5'' = 850 pc. The existence of the double nucleus has been revealed at radio, near-IR (NIR), X-ray, as well as molecular line emission (Norris & Forbes 1995; Kotilainen et al. 1996; Lira et al. 2002; Sakamoto et al. 2006b). Although NGC 3256 is luminous in X-ray (${L}_{0.5-10\mathrm{keV}}\;\sim$ 10⁴² erg s⁻¹; Lira et al. 2002), the presence of an AGN is not supported by a considerable amount of observational evidence (Lípari et al. 2000; Lira et al. 2002; Jenkins et al. 2004). The relatively strong X-ray emission would be stellar in origin since hundreds of young stellar clusters have been found in the central region of this system (Zepf et al. 1999; Alonso-Herrero et al. 2002). Indeed, the above-mentioned X-ray luminosity is consistent with the boundary to separate AGNs from SB galaxies found in deep X-ray surveys (e.g., Alexander et al. 2005). Therefore, we classify this object as an SB galaxy in this paper.

SB galaxy NGC 3628 (${L}_{8-1000\mu {\rm{m}}}$ = 10^10.3L_⊙; Sanders et al. 2003) is a member of the Leo Triplet (Arp 317), including NGC 3627 and NGC 3623 as well. Radio observations of both continuum and hydrogen recombination line emissions revealed that this galaxy shows SB activity in its central ∼500 pc region (Condon et al. 1982; Zhao et al. 1997). A large-scale galactic wind, related to this SB, has been detected in X-ray and Hα emissions (Strickland et al. 2001, 2004). In addition, a subkiloparsec-scale molecular outflow is also found (Tsai et al. 2012). Note that an X-ray point source (${L}_{0.3-8.0\mathrm{keV}}\;\sim$ 10⁴⁰ erg s⁻¹), which is a candidate of an intermediate-mass black hole, was discovered in this galaxy (Strickland et al. 2001). But the location of the object is at least 20'' away from the nucleus. Thus, it would have no contribution to the molecular line observations with the APEX 18'' beam used in this paper (Zhang et al. 2014).

A nearly face-on barred spiral galaxy NGC 7552 (${L}_{8-1000\mu {\rm{m}}}$ = 10^10.9L_⊙; Sanders et al. 2003) has been classified as a LINER galaxy owing to its weak [O i] emission (Durret & Bergeron 1988). It hosts a prominent circumnuclear ring with a radius of ∼25 or 250 pc, which is the powering source of SB activity (e.g., Schinnerer et al. 1997; Brandl et al. 2012). This SB ring is occupied by a number of young stellar clusters (Brandl et al. 2012), which would have formed about 10 Myr ago (Schinnerer et al. 1997). Inside the SB ring, neither X-ray (Liu & Bregman 2005) nor NIR (Forbes et al. 1994) observations have shown clear evidence of nuclear activity. Therefore, this galaxy is genuinely an SB galaxy. High-resolution observations revealed that dense molecular gas is mainly concentrated in the SB ring (Pan et al. 2013).

Unlike other sample galaxies, observational information on this LIRG (${L}_{8-1000\mu {\rm{m}}}$ = 10^11.0L_⊙; Sanders et al. 2003) is currently very limited. However, based on the MIR observations with Spitzer IRS, Alonso-Herrero et al. (2012a) reported that an AGN, if it exists, has almost no contribution to the MIR spectrum. Indeed, [Ne v] 14.32 μm, which is an indicator of the existence of an AGN, was not detected (Alonso-Herrero et al. 2012a). Then, although the observational evidence is lacking, we conclude that an AGN is absent, or at least negligible to the energy budget in this galaxy, and classify it as an SB galaxy.

This is an H ii region in the LMC associated with a clumpy molecular cloud (Seale et al. 2012) with a mass of a few × 10⁵ M_⊙ in total (Wang et al. 2009). The molecular cloud is a site of massive star formation, as evidenced by several embedded young stellar objects (YSOs; e.g., Seale et al. 2009, 2012; Carlson et al. 2012). N113 hosts the most intense H₂O maser of the Magellanic Clouds (e.g., Oliveira et al. 2006) and the OH maser as well (Brooks & Whiteoak 1997). This region (and the LMC as well) is characterized by its low-metallicity ISM (Z ∼ 1/3 Z_⊙; e.g., Hunter et al. 2009). Therefore, this is an appropriate object to investigate the effect of metallicity on molecular chemistry.

This is also the best-studied H ii region in the LMC, close to the evolved SB of 30 Doradus. The N159 complex is a type III giant molecular cloud as classified by Fukui et al. (2008), i.e., a GMC with H ii regions and young star clusters. Indeed, this region is known to host many YSOs, OH maser sources, and ultracompact H ii regions (e.g., Chen et al. 2010). Extensive studies of molecular material have been conducted for this region (e.g., Bolatto et al. 2000; Mizuno et al. 2010; Minamidani et al. 2011; Okada et al. 2015; Paron et al. 2016). The total mass of clumps visible at ¹²CO(3–2) emission is ∼6 × 10⁵ M_⊙ (Minamidani et al. 2008). Recent non-LTE modelings of high-density gas tracers such as HCN, HCO⁺, and CS revealed the existence of warm (T_kin ∼ 80 K) and dense (${n}_{{{\rm{H}}}_{2}}\;\sim$ 5 × 10⁵ cm⁻³) gas (Paron et al. 2016). As is the case of N113 (and LMC), this region is also a site with low metallicity.