## Tremaine-Gunn bound

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The Tremaine-Gunn bound is a lower bound on the mass of dark matter particles, based on the phase-space density evolution. In its simplest and the most robust form it is based on the comparison of the average phase-space density $\bar f$ of dark matter particles in a given object with the phase-space density of degenerate Fermi gas:
Here $m$ is the mass of dark matter particles, $g$ - number of internal degrees of freedom. This bound can be applied to any kind of fermionic dark matter.
A stronger version of this bound (that can be applied also to bosons) is based on the Liouville theorem and holds only when the system obeys Hamiltonian dynamics. The bound compares the maximum of the primordial phase-space density with the maximal value of the phase-space density in a given object today. Unlike the previous case, this bound depends on the primordial distribution of the dark matter.
In realistic astrophysical systems the latter bound can be applied when the influence of the baryonic matter on the dark matter evolution is negligible. This is believed to be (approximately) true for dwarf spheroidal satellites of our Galaxy.
The Tremaine-Gunn bound give $m \gtrsim 400~{\rm eV}$ for any fermionic dark matter.
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