The halo mass function through the cosmic ages
  In this paper we investigate how the halo mass function evolves with
redshift, based on a suite of very large (with N_p = 3072^3 - 6000^3 particles)
cosmological N-body simulations. Our halo catalogue data spans a redshift range
of z = 0-30, allowing us to probe the mass function from the dark ages to the
present. We utilise both the Friends-of-Friends (FOF) and Spherical Overdensity
(SO) halofinding methods to directly compare the mass function derived using
these commonly used halo definitions. The mass function from SO haloes exhibits
a clear evolution with redshift, especially during the recent era of dark
energy dominance (z < 1). We provide a redshift-parameterised fit for the SO
mass function valid for the entire redshift range to within ~20% as well as a
scheme to calculate the mass function for haloes with arbitrary overdensities.
The FOF mass function displays a weaker evolution with redshift. We provide a
`universal' fit for the FOF mass function, fitted to data across the entire
redshift range simultaneously, and observe redshift evolution in our data
versus this fit. The relative evolution of the mass functions derived via the
two methods is compared and we find that the mass functions most closely match
at z=0. The disparity at z=0 between the FOF and SO mass functions resides in
their high mass tails where the collapsed fraction of mass in SO haloes is ~80%
of that in FOF haloes. This difference grows with redshift so that, by z>20,
the SO algorithm finds a ~50-80% lower collapsed fraction in high mass haloes
than does the FOF algorithm, due in part to the significant over-linking
effects known to affect the FOF method.
