  We study the difference between thermally produced fermionic and bosonic hot
dark matter in detail. In the linear regime of structure formation, their
distinct free-streaming behaviours can lead to pronounced differences in the
matter power spectrum. While not detectable with current cosmological data,
such differences will be clearly observable with upcoming large scale weak
lensing surveys for particles as light as m_HDM ~ 0.2 eV. In the nonlinear
regime, bosonic hot dark matter is not subject to the same phase space
constraints that severely limit the amount of fermionic hot dark matter infall
into cold dark matter halos. Consequently, the overdensities in fermionic and
bosonic hot dark matter of equal particle mass can differ by more than a factor
of five in the central part of a halo. However, this unique manifestation of
quantum statistics may prove very difficult to detect unless the mass of the
hot dark matter particle and its decoupling temperature fall within a very
narrow window, 1 < m_HDM/eV < 4 and g* < 30. In this case, hot dark matter
infall may have some observable consequences for the nonlinear power spectrum
and hence the weak lensing convergence power spectrum at l ~ 10^3 - 10^4 at the
percent level.
