  HCN and CO line diagnostics provide new insight into the OH megamaser (OHM)
phenomenon, suggesting a dense gas trigger for OHMs. We identify three physical
properties that differentiate OHM hosts from other starburst galaxies: (1) OHMs
have the highest mean molecular gas densities among starburst galaxies; nearly
all OHM hosts have <n(H2)> = 10^3-10^4 cm^-3 (OH line-emitting clouds likely
have n(H2) > 10^4 cm^-3). (2) OHM hosts are a distinct population in the
nonlinear part of the IR-CO relation. (3) OHM hosts have exceptionally high
dense molecular gas fractions, L(HCN)/L(CO)>0.07, and comprise roughly half of
this unusual population. OH absorbers and kilomasers generally follow the
linear IR-CO relation and are uniformly distributed in dense gas fraction and
L(HCN), demonstrating that OHMs are independent of OH abundance. The fraction
of non-OHMs with high mean densities and high dense gas fractions constrains
beaming to be a minor effect: OHM emission solid angle must exceed 2 pi
steradians. Contrary to conventional wisdom, IR luminosity does not dictate OHM
formation; both star formation and OHM activity are consequences of tidal
density enhancements accompanying galaxy interactions. The OHM fraction in
starbursts is likely due to the fraction of mergers experiencing a temporal
spike in tidally driven density enhancement. OHMs are thus signposts marking
the most intense, compact, and unusual modes of star formation in the local
universe. Future high redshift OHM surveys can now be interpreted in a star
formation and galaxy evolution context, indicating both the merging rate of
galaxies and the burst contribution to star formation.
