Analytic fluid approximation for warm dark matter

Abstract

We present the cosmological evolution of the velocity of a massive particle, along with its equation of state. Both quantities are expressed in terms of $a_{nr}$, the moment when a massive particle becomes nonrelativistic. The expressions for the energy density and pressure for the background evolution are also in terms of $a_{nr}$; therefore, the perturbation equations for any massively decoupled particle, i.e., warm dark matter (WDM) or neutrinos, can be computed in the fluid approximation. A relation between the mass of the WDM particle, $m_{\mathrm{wdm}}$, with the moment of the nonrelativistic transition, $a_{nr}$, and the temperature ratio with the neutrinos, $T_{\mathrm{wdm}}/T_{\nu }$, are obtained and compared to results from Boltzmann solvers as CLASS with a non-cold relic as WDM. We found that using the analytic fluid approximation, the value of $a_{nr}$ is 2.1% different on average in a wide range of WDM masses, and the temperature ratio is 7.1% lower than previous parametrizations. A smooth velocity dispersion for the WDM allows us to compute the cut-off scale in structure formation due to the free-streaming (${\lambda }_{fs}$), which is a feature for a WDM particle to explain the satellite problem. The cut-off in the matter power spectrum and halo mass function using the analytic fluid approximation is similar to the Boltzmann solvers with a non-cold relic and the transfer function from numerical simulations. This approach provides a more detailed description and a deeper understanding of the WDM cosmological evolution by understanding the velocity dispersion of a WDM particle. Comprehensive numerical modeling can incorporate the analytical fluid formulation, potentially improving calculation performance, for instance, running MCMC for a $\Lambda$WDM scenario using CMB Planck and WiggleZ data, we obtained a lower bound for the WDM mass $m_{\mathrm{wdm}}>70.3\mathrm{eV}$ at $1\sigma$ confidence. Still, more data at small scales or a combination with other observations are needed to constrain the mass value of the WDM particle.