Free streaming in mixed dark matter
Daniel Boyanovsky

Free streaming in a \emph{mixture} of collisionless non-relativistic dark matter (DM) particles is studied by solving the linearized Vlasov equation implementing methods from the theory of multicomponent plasmas. The mixture includes Fermionic, condensed and non-condensed Bosonic particles decoupling in equilibrium while relativistic, heavy thermal relics that decoupled when non-relativistic (WIMPs), and sterile neutrinos that decouple \emph{out of equilibrium} when they are relativistic. The different components interact via the self-consistent gravitational potential that they source. The free-streaming length $\lambda_{fs}$ is obtained from the marginal zero of the gravitational polarization function, which separates short wavelength Landau-damped from long wavelength Jeans-unstable \emph{collective} modes. At redshift $z$ we find $ \frac{1}{\lambda^2_{fs}(z)} = \frac{1}{(1+z)}\, \big[ \frac{0.071}{\textrm{kpc}}\big]^2 \sum_{a} \nu_a \, g^\frac{2}{3}_{d,a}\,\left( {m_a}/{\mathrm{keV}}\right)^2 I_a $, where $0\leq \nu_a \leq 1$ are the \emph{fractions} of the respective DM components of mass $m_a$ that decouple when the effective number of ultrarelativistic degrees of freedom is $g_{d,a}$, and $I_a$ are dimensionless ratios of integrals of the distribution functions which only depend on the microphysics at decoupling and are obtained explicitly in all the cases considered. If sterile neutrinos produced either resonantly or non-resonantly that decouple near the QCD scale are the \emph{only} DM component, we find $\lambda_{fs}(0) \simeq 7\,\mathrm{kpc} \,(\mathrm{keV}/m)$ (non-resonant), $\lambda_{fs}(0) \simeq 1.73\,\mathrm{kpc}\, (\mathrm{keV}/m)$ (resonant). If WIMPs with $m_{wimp} \gtrsim 100\,\mathrm{GeV}$ decoupling at $T_d \gtrsim 10 \,\mathrm{MeV}$ are present in the mixture with $\nu_{wimp} \gg 10^{-12}$, $\lambda_{fs}(0) \lesssim 6.5 \times 10^{-3}\,\mathrm{pc}$ is \emph{dominated} by CDM. If a Bose Einstein condensate is a DM component its free streaming length is consistent with CDM because of the infrared enhancement of the distribution function.

© 2008 The American Physical Society.