**3.1. Tremaine-Gunn bound**

Before treating the phase space evolution, we discuss another important consequence of finite neutrino thermal speed: high-speed neutrinos cannot be tightly packed into galaxy halos. This fact can be used to place a lower bound on the neutrino mass if neutrinos make up the dark matter in galaxy halos (Tremaine & Gunn 1979).

The initial phase space density for massive neutrinos is a relativistic Fermi-Dirac distribution (preserved from the time when the neutrinos decoupled in the early universe):

(3.4) |

where ** p** is the comoving canonical momentum of eq. (1.13),

Tremaine & Gunn (1979)
noted that because of phase mixing (discussed
further below), the maximum coarse-grained phase space density of
massive neutrinos today is less than the maximum of
*f*_{0}(** p**),

Although the neutrino mass bound is somewhat model-dependent
because the actual coarse-grained distribution in galactic halos is
unknown, we can get a reasonable estimate by assuming an isothermal
sphere: a Maxwell-Boltzmann distribution with constant velocity
dispersion ^{2}
(at *a* = 1 so that there is no distinction between
proper and comoving):

(3.5) |

In a self-gravitating system there are a family of spherical density
profiles
(*r*) =
*m*_{}
*n*(*r*) obeying hydrostatic equilibrium:

(3.6) |

The simplest case is the singular isothermal sphere with
*r*^{-2}; the reader can easily check that
=
^{2} /
(2 *Gr*^{2}).
Imposing the phase space bound at radius *r* then gives

(3.7) |

Up to overall numerical factors, this is the Tremaine-Gunn bound.

The singular isothermal sphere is probably a good model where
the rotation curve produced by the dark matter halo is flat, but
certainly breaks down at small radius. Because the neutrino mass
bound is stronger for smaller
*r*^{2},
the uncertainty in the
halo core radius (interior to which the mass density saturates) limits
the reliability of the neutrino mass bound.

For the Local Group dwarf galaxies in Draco and Ursa Minor, measurements
of stellar velocity dispersions suggest
is a few to about 10
km s^{-1}
(Pryor & Kormendy 1990).
If these galaxies have isothermal
halos at *r* = 1 kpc, the crude bound of eq. (3.7)
implies *m*_{} is
greater than a few eV.