We present the results of CO(J=3-2) on-the-fly mappings of two nearby non-barred spiral galaxies NGC 628 and NGC 7793 with the Atacama Submillimeter Telescope Experiment at an effective angular resolution of 25. We successfully obtained global distributions of CO(J=3-2) emission over the entire disks at a sub-kpc resolution for both galaxies. We examined the spatially-resolved (sub-kpc) relationship between CO(J=3-2) luminosities (LCO(3-2)) and infrared (IR) luminosities (LIR) for NGC 628, NGC 7793, and M 83, and compared it with global luminosities of JCMT Nearby Galaxy Legacy Survey sample. We found a striking linear LCO(3-2)-LIR correlation over the 4 orders of magnitude, and the correlation is consistent even with that for ultraluminous infrared galaxies and submillimeter selected galaxies. In addition, we examined the spatially-resolved relationship between CO(J=3-2) intensities (ICO(3-2)) and extinction-corrected star formation rates (SFRs) for NGC 628, NGC 7793, and M 83, and compared it with that for GMCs in M 33 and 14 nearby galaxy centers. We found a linear ICO(3-2)-SFR correlation with 1 dex scatter. We conclude that the CO(J=3-2) star formation law (i.e., linear LCO(3-2)-LIR and ICO(3-2)-SFR correlations) is universally applicable to various types and spatial scales of galaxies, from spatially-resolved nearby galaxy disks to distant IR-luminous galaxies, within 1 dex scatter.
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