Scaling relations between the total infrared luminosities and the line luminosities of dense-gas tracers. The positions of the GEMS (red points) are compared with the low-redshift relations inferred for a diverse sample of normal blue star-forming galaxies, starbursts, and ULIRGs, and some AGN-dominated sources (black squares, Zhang et al. 2014). Solid and dashed-dotted black lines show their best-fit relations and the ±1σ scatter, respectively. The relations obtained in Tan et al. (2018) for six nearby star-forming galaxies resolved at sub-kpc (gray points) appear as gray dotted lines. The HCN and HCO⁺ luminosities of the GEMS are plotted for J = 4–3, after correcting L_IR for μ_dust and for μ_gas, the magnification factors of the dust and gas components listed in C18. For PLCK_G244.8+54.9, we plot the measured J = 4–3 line luminosities, while for PLCK_G092.5+42.9 and PLCK_G145.2+50.9, we convert the J = 5–4 line luminosities to J = 4–3 using measurements in the circumnuclear disk near the Galactic center (see text, Mills et al. 2013). Previous detections and upper limits of high-redshift sources are plotted as filled symbols and include dusty star-forming galaxies from a subset of the SPT sample (orange circles, Béthermin et al. 2018), the stacking analysis of SPT sources (orange stars, Spilker et al. 2014), four H-ATLAS lensed SMGs (SDP9 and SDP11 as orange triangles, Oteo et al. 2017 and Yang 2017, NCv1.143 and G09v1.97 as upside-down triangles), and the Cosmic Eyelash (orange squares, Danielson et al. 2011). We also show the Cloverleaf QSO (green circle, Riechers et al. 2011). Measurements taken from the literature for J = 5–4 or J = 3–2 were converted to J = 4–3 in the same way as for the GEMS, and strongly lensed sources were corrected for the magnification factors.