  The mapping between dark matter halo mass, galaxy stellar mass, and galaxy
cold gas mass is not a simple linear relation, but is influenced by a wide
array of galaxy formation processes. We implement observationally-normalized
relations between dark matter halo mass, stellar mass, and cold gas mass to
explore these mappings, with specific emphasis on the correlation between
different definitions of a major galaxy merger. We always define a major merger
by a mass ratio m/M>0.3, but allow the masses used to compute this ratio to be
defined in one of three ways: dark matter halo masses, galaxy stellar masses,
or galaxy baryonic masses (stars and cold gas). We find that the merger ratio
assigned to any particular merger event depends strongly on which of these
masses is used, with the mapping between different mass ratio definitions
showing strong evolution with halo mass and redshift. For example, major dark
matter mergers (>0.3) in small galaxies (M_DM < 10^11 Msun) typically
correspond to very minor stellar mergers (<1/20). These mergers contain
significant dark matter mass, and should cause noticable morphological
disruption to the primary galaxy, even though there is no observable bright
companion. In massive galaxies, there is an opposite effect, with bright
companion galaxies corresponding to only minor dark matter mergers. We
emphasize that great care must be taken when comparing mergers based on
different mass ratio definitions.
