One of the most beautiful ideas about excluding the possibility of neutrinos to be the dark matter candidates is through a simple answer to a simple question. What is the number of particles that could be in a galaxy?. In 1979 there was a paper published trying to answer this question (1). It is called Tremaine-Gunn bound, which put an also constrain for the dark matter candidates.

We know that the galaxy is a pretty big object, and particles are microscopic objects, so the answer is definitely “a lot.” But in physics, there are two types of particles, fermions (the matter particles we composed of) and bosons. For fermions, however, the answer isn’t “infinity.” And that turns out to be very interesting. Fermions are spin-1/2 particles, and they have the property that two fermions cannot exist in the same quantum state. This is the Pauli Exclusion Principle.

Quantum states include “position,” so what this means is that, if you start trying to pile fermions on top of each other, they will have to sit “next” to each other. This is why all the fields we think of as “matter” are fermionic (electrons, protons, neutrons). So let’s imagine that dark matter is some unknown fermion. Then we can start piling up dark matter inside a galaxy. Each particle you add to the galaxy makes it’s mass a bit larger, but also takes up a slot in the position and momentum quantum states. Eventually, you’ll have added so many fermions that the only quantum states that remain open would require a fermion to be so far away from the galactic center and moving so fast that it would no longer be gravitationally bound to the galaxy.

So there is a maximum number of dark matter fermions you can add to a galaxy. If the dark matter is too light, then this number of fermions would not possibly accommodate the total dark matter mass of the galaxy. Looking at dwarf galaxies, you can use this technique to place the simplest version of the Tremaine-Gunn bound: if dark matter is a fermion, it needs to be heavier than about one keV. You can sharpen this argument at bit using phase-space mixing and get the limit up to ~ three keV.
Interestingly, despite neutrinos being otherwise ideal as a dark matter candidate, this limit told us they could not be dark matter. We know that neutrinos are much lighter than 1-3 keV. So you literally cannot cram enough neutrinos into a galaxy for them to be the dark matter. Amusingly, this Pauli Exclusion Limit for fermions is also what supports a white dwarf or neutron star from gravitational collapse. If the star got smaller, there wouldn’t be enough room for the electrons (dwarfs) or neutrons (neutron stars) without doubling up on quantum states. If dark matter is a boson, you can’t use the Tremaine-Gunn bound, since bosons are okay with all sitting in the same quantum state. So in principle, you can fit an infinite number inside a galaxy. You can use another property of quantum fields to constrain bosonic dark matter, though. If the particle is too light, the quantum wavefunction is so dispersed it cannot be localized inside the galaxy.

This puts a lower bound of ~10^-22 eV on bosonic dark matter. Rather less than the bound for fermionic dark matter. But at least this is even a “thing.” Of course, there are other reasons to eliminate neutrinos as a dark matter candidate, but yet we know more about what couldn’t be a dark matter, not what is the dark matter. 😏

[1] (S. Tremaine and J. E. Gunn, Phys. Rev. Lett. 42, 407 (1979).)