Observations of dense molecular gas in a quasar host galaxy at z=6.42:: Further evidence for a nonlinear dense gas-star formation relation at early cosmic times

We report a sensitive search for the HCN(J = 2 -> 1) emission line toward SDSS J114816.64 + 525150.3 (hereafter J1148 + 5251) at z = 6.42 with the Very Large Array (VLA). HCN emission is a star formation indicator, tracing dense molecular hydrogen gas [nH(2) >= 10(4) cm(-3)] within star-forming molecular clouds. No emission was detected in the deep interferometer maps of J1148 + 5251. We derive a limit for the HCN line luminosity of L'(HCN)<3.3 x 10(9) K km s(-1) pc(2), corresponding to a HCN/CO luminosity ratio of L'(HCN)/L'(CO) < 0.13. This limit is consistent with a fraction of dense molecular gas in J1148 + 5251 within the range of nearby ultraluminous infrared galaxies (ULIRGs; median value: L'(HCN)/L'(CO) =0.17(-0.08)(+0.05)) and HCN- detected z > 2 galaxies (0.17(-.008)(+0.009)). The relationship between L'(HCN) and L-FIR is considered to be a measure of the efficiency at which stars form out of dense gas. In the nearby universe, these quantities show a linear correlation, and thus, a practically constant average ratio. In J1148 + 5251, we find L-FIR/L'(HCN) > 6600. This is significantly higher than the average ratios for normal nearby spiral galaxies (L-FIR/L'(HCN) =580(-270)(+580)) and ULIRGs (740), but consistent with a rising trend as indicated by other galaxies (predominantly quasars;1525(-475)(+1300)). It is unlikely that this rising trend can be accounted for by a contribution of active galactic nucleus (AGN) heating to alone, and may hint at a higher median gas density and/ or elevated star formation efficiency toward the more luminous high- redshift systems. There is marginal evidence that the L-FIR/L'(HCN) ratio in J1148 + 5251 may even exceed the rising trend set by other z>2 galaxies; however, only future facilities with very large collecting areas such as the Square Kilometer Array (SKA) will offer the sensitivity required to further investigate this question.