Analytic Fluid Approximation for Warm Dark Matter
  We present the full evolution of the velocity of a massive particle, along
with the equation of state we can compute the energy density and pressure
evolution for the background evolution. It is also natural to compute the
perturbation equations for any massive decoupled particle, i.e. warm dark
matter (WDM) or neutrinos, in the fluid approximation. Using this approach we
analytically compute the time when the WDM stop being relativistic, $a_{nr}$,
which is 2.6\% different respect to the exact Boltzmann solution. Using the
fluid approximation the matter power spectrum is computed faster and with great
accuracy, the cut-off in structure formation due to the free-streaming
($\lambda_{fs}$) of the particle, characteristic for a WDM particle, is
replicated in both matter power spectrum and halo mass function. With this
approach, we have a deeper understanding of the WDM physics that lead us to
show that the temperature the dark matter can be computed as a function of
known properties of the WDM particle. This formulation can be integrated into
comprehensive numerical modeling reasonable increasing the performance in the
calculations, therefore, we analyze the parameter $a_{nr}$ in a $\Lambda$WDM
model using CMB Planck data combined with matter power spectrum data set of
WiggleZ, obtaining a lower bound for the WDM mass $m_{\rm wdm} = 70.3$ eV at
86\% confidence, this value is consistent with WiggleZ data set but more data
at small scales or a combination with other observations are needed to stronger
constrain the mass value of the WDM particle.
