The universality of the virial halo mass function and models for
  non-universality of other halo definitions
  The abundance of galaxy clusters can constrain both the geometry and growth
of structure in our Universe. However, this probe could be significantly
complicated by recent claims of nonuniversality -- non-trivial dependences with
respect to the cosmological model and redshift. In this work we analyse the
dependance of the mass function on the way haloes are identified and establish
if this can cause departures from universality. In order to explore this
dependance, we use a set of different N-body cosmological simulations (Le
SBARBINE simulations), with the latest cosmological parameters from the Planck
collaboration; this first suite of simulations is followed by a lower
resolution set, carried out with different cosmological parameters. We identify
dark matter haloes using a Spherical Overdensity algorithm with varying
overdensity thresholds (virial, 2000rho_c, 1000rho_c, 500rho_c, 200rho_c and
200rho_b) at all redshifts. We notice that, when expressed in term of the
rescaled variable nu, the mass functionfor virial haloes is a nearly universal
as a function of redshift and cosmology, while this is clearly not the case for
the other overdensities we considered. We provide fitting functions for the
halo mass function parameters as a function of overdensity, that allow to
predict, to within a few percent accuracy, the halo mass function for a wide
range of halo definitions, redshifts and cosmological models. We then show how
the departures from universality associated with other halo definitions can be
derived by combining the universality of the virial definition with the
expected shape of the density profile of halos.
