Relation of the intrinsic H₂O luminosity with the infrared luminosity for the seven ultra-luminous infrared galaxies reported in this paper, other well-known high-z sources taken from the literature, and local ULIRGs (L_IR < 4 × 10¹² L_⊙, Yang et al., in prep.). The luminosity is reported to the H₂O (2₁₁ − 2₀₂, 752.03 GHz) emission line (referred to as “line 1” in this paper), where we adopted an Arp 220 H₂O line ratio when another H₂O line (“line 2”) was measured (see text). The solid line shows a best-fit power-law, , where α = 1.15 ± 0.10. Error bars are defined in the following way: i) for L_IR, they combine the uncertainties on the values of μ L_IR and the uncertainties on μ given in Table 2; ii) for L_H2O, they include the errors on I_H2O of Table 4, and the same uncertainties on μ except that when the latter is <20% it is replaced by 20% (see Sect. 4.1). Because of their peculiarities, values (in blue) for SDP.81 and G12.v2.30 (both underestimated by factors possibly up to ~1.5−2 because they are very extended, see text), and those of APM 08279+5255 and Mrk 231 (type 1 and type 2 QSOs, respectively) are not taken into account for the fit.