HCN luminosity is a tracer of dense molecular gas, n(H₂) 3 × 10⁴ cm^-3, associated with star-forming giant molecular cloud (GMC) cores. We present the results and analysis of our survey of HCN emission from 65 infrared galaxies, including nine ultraluminous infrared galaxies (ULIGs, L_IR 10¹² L), 22 luminous infrared galaxies (LIGs, 10¹¹ L < L_IR 10¹² L), and 34 normal spiral galaxies with lower IR luminosity (most are large spiral galaxies). We have measured the global HCN line luminosity, and the observations are reported in Paper I. This paper analyzes the relationships between the total far-IR luminosity (a tracer of the star formation rate), the global HCN line luminosity (a measure of the total dense molecular gas content), and the CO luminosity (a measure of the total molecular content). We find a tight linear correlation between the IR and HCN luminosities L_IR and L_HCN (in the log-log plot) with a correlation coefficient R = 0.94, and an almost constant average ratio L_IR/L_HCN = 900 L (K km s^-1 pc²)^-1. The IR-HCN linear correlation is valid over 3 orders of magnitude including ULIGs, the most luminous objects in the local universe. The direct consequence of the linear IR-HCN correlation is that the star formation law in terms of dense molecular gas content has a power-law index of 1.0. The global star formation rate is linearly proportional to the mass of dense molecular gas in normal spiral galaxies, LIGs, and ULIGs. This is strong evidence in favor of star formation as the power source in ultraluminous galaxies since the star formation in these galaxies appears to be normal and expected given their high mass of dense star-forming molecular gas.

The HCN-CO correlation is also much tighter than the IR-CO correlation. We suggest that the nonlinear correlation between L_IR and L_CO may be a consequence of the stronger and perhaps more physical correlations between L_IR and L_HCN and between L_HCN and L_CO. Thus, the star formation rate indicated by L_IR depends on the amount of dense molecular gas traced by HCN emission, not the total molecular gas traced by CO emission. One of the main arguments in favor of an active galactic nucleus (AGN) as the power source in ULIGs is the anomalously high ratio L_IR/L_CO or L_IR/M(H₂) or high star formation rate per M of gas, compared with that from normal spiral galaxies. This has been interpreted as indicating that a dust-enshrouded AGN is required to produce the very high luminosity. Viewed in terms of the dense gas mass the situation is completely different. The ratio L_IR/L_HCN or L_IR/M_dense, a measure of the star formation rate per solar mass of dense gas, is essentially the same in all galaxies including ULIGs. The ratio L_IR/M_dense is virtually independent of galaxy luminosity and on average L_IR/M_dense ≈ 90L/M, about the same as in GMC cores but much higher than in GMCs. We find that ULIGs simply have a large quantity of dense molecular gas and thus produce a prodigious starburst that heats the dust, produces the IR, and blocks all or most optical radiation. The HCN global luminosity may be used as an indicator of the star formation rate in high-redshift objects including hyperluminous galaxies.

The HCN/CO ratio is an indicator of the dense molecular gas fraction and gauges the globally averaged molecular gas density. We find that the HCN/CO ratio is a powerful starburst indicator. All galaxies in our sample with a high dense gas mass fraction indicated by L_HCN/L_CO > 0.06 are LIGs or ULIGs. Normal spirals all have similar and low dense gas fractions L_HCN/L_CO = 0.02-0.05. The global star formation efficiency depends on the fraction of the molecular gas in a dense phase.