  The behavior of a decoupled ideal Fermi gas in a homogeneously expanding
three-dimensional volume is investigated, starting from an equilibrium
spectrum. In case the gas is massless and/or completely degenerate, the
spectrum of the gas can be described by an effective temperature and/or an
effective chemical potential, both of which scale down with the volume
expansion. In contrast, the spectrum of a decoupled massive and non-degenerate
gas can only be described by an effective temperature if there are strong
enough self-interactions such as to maintain an equilibrium distribution.
Assuming perpetual equilibration, we study a decoupled gas which is
relativistic at decoupling and then is red-shifted until it becomes
non-relativistic. We find expressions for the effective temperature and
effective chemical potential which allow us to calculate the final spectrum for
arbitrary initial conditions. This calculation is enabled by a new expansion of
the Fermi-Dirac integral, which is for our purpose superior to the well-known
Sommerfeld expansion. We also compute the behavior of the phase space density
under expansion and compare it to the case of real temperature and real
chemical potential. Using our results for the degenerate case, we also obtain
the mean relic velocity of the recently proposed non-thermal cosmic neutrino
background.
