The nature of dark matter is still unknown, but in this entry we will show that it cannot be made up of low mass Fermions. This is because the exclusion principle limits the number of such particles in a given phase space, so if the mass of the individual particle is below a certain threshold, the total mass will be too low to account for the necessary gravitational field.
Let us consider a galaxy with radius and velocity dispersion . The virial mass is
If the mass of a single dark matter particle is , then the phase space volume is
and the maximum number of particles that can fit into this volume is (up to a factor of a few, due to spin)
So the upper bound on the total mass of all the dark matter particles is
The upper bound on the total mass has to exceed the virial mass, and the corresponding condition on the mass of a dark matter particle is
(substitution here). This limit excludes neutrinos from being dark matter particles, since their rest mass energy is lower than 1 eV.