  We report a sensitive search for the HCN(J=2-1) emission line towards SDSS
J1148+5251 at z=6.42 with the VLA. HCN emission is a star formation indicator,
tracing dense molecular hydrogen gas (n(H2) >= 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 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 {+0.05/-0.08}) and HCN-detected z>2
galaxies (0.17 {+0.09/-0.08}). The relationship between L'(HCN) and L(FIR) is
considered to be a measure for 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 {+510/-270}) and ULIRGs
(740 {+505/-50}), but consistent with a rising trend as indicated by other z>2
galaxies (predominantly quasars; 1525 {+1300/-475}). It is unlikely that this
rising trend can be accounted for by a contribution of AGN heating to L(FIR)
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 SKA will offer the
sensitivity required to further investigate this question.
