Modeled L_H2O/L_IR × (100 km s^-1/ ΔV) for lines 2 (upper), 4 (middle), and 6 (lower) as a function of the f₂₅/f₆₀ color. The dashed vertical lines indicate the lower and upper limits for f₂₅/f₆₀ measured by Y13. In the upper panel, the small numbers below the squares indicate the value of T_dust , and τ₁₀₀ is also indicated. Blue squares: N_H2O/ ΔV = 10¹⁵cm^-2/ (km s^-1) and τ₁₀₀ = 0.1; magenta: same as blue symbols but with collisional excitation included with T_gas = 150 K and n_H2 = 3 × 10⁵ cm^-3; green: N_H2O/ΔV = 10¹⁵cm^-2/ (km s^-1) and τ₁₀₀ = 1.0; black: same as green but with N_H2O/ΔV = 5 × 10¹⁵cm^-2/ (km s^-1). Red squares show results for fixed T_dust = 55 and 65 K with N_H2O/(ΔVτ₁₀₀) = 5 × 10¹⁵cm^-2/ (km s^-1) and τ₁₀₀ = 0.1−0.3. The starred symbols indicate the positions of Arp 220 (blue), NGC 6240 (green), Mrk 231 (red), and NGC 1068 (magenta), as reported by Rangwala et al. (2011), Meijerink et al. (2013), G-A10, and Y13, respectively. When compared with observations, the modeled L_IR values should be considered a fraction of the observed IR luminosities (Sect. 4.3.1). If f₆₀ is contaminated by cold dust, the points would move to the left.