Is it possible to tell the difference between fermionic and bosonic hot dark matter?

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.