We present deep IRAM Plateau de Bure Interferometer (PdBI) observations, searching for CO-emission toward two massive, non-lensed Lyman break galaxies (LBGs) at z=3.216 and 4.058. With one low significance CO detection (3.5 sigma) and one sensitive upper limit, we find that the CO lines are >~ 3-4 times weaker than expected based on the relation between IR and CO luminosities followed by similarly, massive galaxies at z=0-2.5. This is consistent with a scenario in which these galaxies have low metallicity, causing an increased CO-to-H_2 conversion factor, i.e., weaker CO-emission for a given molecular (H_2) mass. The required metallicities at z>3 are lower than predicted by the fundamental metallicity relation (FMR) at these redshifts, consistent with independent evidence. Unless our galaxies are atypical in this respect, detecting molecular gas in normal galaxies at z>3 may thus remain challenging even with ALMA.
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