**3.2. Phase Space Constraint**

We have just seen that light neutrinos must satisfy an upper bound on the sum of their masses. But now we will discuss a lower bound on neutrino mass that arises because they must be rather massive to form the dark matter in galaxies, since their phase space density is limited by the Pauli exclusion principle. A slightly stronger bound follows from the fact that they were not degenerate in the early universe.

The phase space constraint [36] follows from Jeans's theorem in classical mechanics to the effect that the maximum 6-dimensional phase space density cannot increase as a system of collisionless particles evolves. At early times, before density inhomogenitites become nonlinear, the neutrino phase space density is given by the Fermi-Dirac distribution

(5) |

where here *h* is Planck's constant and
*g*_{} = 2 for each
species of left-handed plus
right-handed .
Since momentum and temperature both scale as redshift *z* as the
universe expands, this distribution remains valid after neutrinos
drop out of thermal equilibrium at ~ 1 MeV, and even into the
nonrelativistic regime
*T*_{}
< *m*_{}
[28].
The standard version of the
phase space constraint follows from demanding that
the central phase space density
9[2 (2)^{5/2}
*G r*_{c}^{2}
*m*_{}^{4}]^{-1} of the DM halo, assumed to be
an isothermal sphere of core radius *r*_{c} and one-dimensional
velocity dispersion , not
exceed the maximum value of the initial phase space density
*n*_{}(0) =
*g*_{} / 2*h*^{3}.
The result is

(6) |

The strongest lower limits on
*m*_{} follow from
applying this to the smallest galaxies.
Both theoretical arguments regarding the dwarf spheroidal
(*dS*) satellite galaxies of the Milky Way
[37]
and data on Draco, Carina, and Ursa Minor made it clear some time ago that
dark matter dominates the gravitational potential of
these *dS* galaxies, and the case has only strengthened with
time [38].
The phase space constraint then sets a lower limit
[39]
*m*_{} > 500 eV, which
is completely incompatible with the
cosmological constraint eq. (4). However, this
argument only excludes neutrinos as the DM in certain small
galaxies; it remains possible that the DM in these galaxies
is (say) baryonic, while that in larger galaxies such as our
own is (at least partly) light neutrinos.
A more conservative phase space constraint was obtained for the Draco
and Ursa Minor dwarf spheroidals
[40], but the
authors concluded that neutrinos consistent with the cosmological upper
bound on *m*_{} cannot
be the DM in these galaxies. A similar
analysis applied to the gas-rich low-rotation-velocity dwarf irregular
galaxy DDO 154
[41] gave a limit
*m*_{} > 94 eV, again
inconsistent with the cosmological upper bound.