Modelling the Effects of Dark Matter Substructure on Globular Cluster
  Evolution with the Tidal Approximation
  We present direct $N$-body simulations of tidally filling 30,000 ${\rm
M}_\odot$ star clusters orbiting between 10 kpc and 100 kpc in galaxies with a
range of dark matter substructure properties. The time-dependent tidal force is
determined based on the combined tidal tensor of the galaxy's smooth and clumpy
dark matter components, the latter of which causes fluctuations in the tidal
field that can heat clusters. The strength and duration of these fluctuations
are sensitive to the local dark matter density, substructure fraction, sub-halo
mass function, and the sub-halo mass-size relation. Based on the cold dark
matter framework, we initially assume sub-halos are Hernquist spheres following
a power-law mass function between $10^5$ and $10^{11} {\rm M}_\odot$ and find
that tidal fluctuations are too weak and too short to affect star cluster
evolution. Treating sub-halos as point masses, to explore how denser sub-halos
affect clusters, we find that only sub-halos with masses greater than $10^{6}
{\rm M}_\odot$ will cause cluster dissolution times to decrease. These
interactions can also decrease the size of a cluster while increasing the
velocity dispersion and tangential anisotropy in the outer regions via tidal
heating. Hence increased fluctuations in the tidal tensor, especially
fluctuations that are due to low-mass halos, do not necessarily translate into
mass loss. We further conclude that the tidal approximation can be used to
model cluster evolution in the tidal fields of cosmological simulations with a
minimum cold dark matter sub-halo mass of $10^{6} {\rm M}_\odot$, as the effect
of lower-mass sub-halos on star clusters is negligible.
