  The mass function of dark matter halos is one of the most fundamental
statistics in structure formation. Many theoretical models (such as
Press-Schechter theory) are based on the notion that it could be universal,
meaning independent of redshift and cosmology, when expressed in the
appropriate variables. However, simulations exhibit persistent
non-universalities in the mass functions of the virial mass and other commonly
used spherical overdensity definitions. We systematically study the
universality of mass functions over a wide range of mass definitions, for the
first time including the recently proposed splashback mass, Msp. We confirm
that, in LambdaCDM cosmologies, all mass definitions exhibit varying levels of
non-universality that increase with peak height and reach between 20% and 500%
at the highest masses we can test. Mvir, M200m, and Msp exhibit similar levels
of non-universality. There are, however, two regimes where the splashback mass
functions are significantly more universal. First, they are universal to 10% at
z<2, whereas spherical overdensity definitions experience an evolution due to
dark energy. Second, when additionally considering self-similar cosmologies
with extreme power spectra, splashback mass functions are remarkably universal
(to between 40% and 60%) whereas their spherical overdensity counterparts reach
non-universalities between 180% and 450%. These results strongly support the
notion that the splashback radius is a physically motivated definition of the
halo boundary. We present a simple, universal fitting formula for splashback
mass functions that accurately reproduces our simulation data.
