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.