The phase space density of fermionic dark matter haloes
  We have performed a series of numerical experiments to investigate how the
primordial thermal velocities of fermionic dark matter particles affect the
physical and phase space density profiles of the dark matter haloes into which
they collect. The initial particle velocities induce central cores in both
profiles, which can be understood in the framework of phase space density
theory. We find that the maximum coarse-grained phase space density of the
simulated haloes (computed in 6 dimensional phase space using the EnBid code)
is very close to the theoretical fine-grained upper bound, while the pseudo
phase space density, Q ~ {\rho}/{\sigma}^3, overestimates the maximum phase
space density by up to an order of magnitude. The density in the inner regions
of the simulated haloes is well described by a 'pseudo-isothermal' profile with
a core. We have developed a simple model based on this profile which, given the
observed surface brightness profile of a galaxy and its central velocity
dispersion, accurately predicts its central phase space density. Applying this
model to the dwarf spheroidal satellites of the Milky Way yields values close
to 0.5 keV for the mass of a hypothetical thermal warm dark matter particle,
assuming the satellite haloes have cores produced by warm dark matter free
streaming. Such a small value is in conflict with the lower limit of 1.2 keV
set by observations of the Lyman-{\alpha} forest. Thus, if the Milky Way dwarf
spheroidal satellites have cores, these are likely due to baryonic processes
associated with the forming galaxy, perhaps of the kind proposed by Navarro,
Eke and Frenk and seen in recent simulations of galaxy formation in the cold
dark matter model.
