Massive Neutrinos in Cosmology: Analytic Solutions and Fluid
  Approximation
  We study the evolution of linear density fluctuations of free-streaming
massive neutrinos at redshift of z<1000, with an explicit justification on the
use of a fluid approximation. We solve the collisionless Boltzmann equation in
an Einstein de-Sitter (EdS) universe, truncating the Boltzmann hierarchy at
lmax=1 and 2, and compare the resulting density contrast of neutrinos,
\delta_{\nu}^{fluid}, with that of the exact solutions of the Boltzmann
equation that we derive in this paper. Roughly speaking, the fluid
approximation is accurate if neutrinos were already non-relativistic when the
neutrino density fluctuation of a given wavenumber entered the horizon. We find
that the fluid approximation is accurate at few to 25% for massive neutrinos
with 0.05<m_{\nu}<0.5eV at the scale of k<0.4~hMpc^{-1} and redshift of z<10.
This result quantifies the limitation of the fluid approximation, for the
massive neutrinos with m_{\nu}<0.5eV. We also find that the density contrast
calculated from fluid equations (i.e., continuity and Euler equations) becomes
a better approximation at a lower redshift, and the accuracy can be further
improved by including an anisotropic stress term in the Euler equation. The
anisotropic stress term effectively increases the pressure term by a factor of
9/5.
