Free streaming in mixed dark matter

Free streaming in a \emph{mixture} of collisionless non-relativistic dark matter (DM) particles is studied by implementing methods from the theory of multicomponent plasmas. The mixture includes Fermionic, condensed and non condensed Bosonic particles decoupling in equilibrium while relativistic, heavy non-relativistic thermal relics (WIMPs), and sterile neutrinos that decouple \emph{out of equilibrium} when they are relativistic. 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}{$\text{kpc}$} \big]^2 \sum_{a}\nu_a g^{2/3}{d,a}({m_a}/{$\mathrm{k}\mathrm{e}\mathrm{V}$})^2 I_a $,where$0\leq \nu_a \leq 1$arethe\mathit{\text{fractions}}oftherespectiveDMcomponentsofmass$m_a$ that decouple when the effective number of ultrarelativistic degrees of freedom is $g_{d,a}$, and $I_a$ only depend on the distribution functions at decoupling, given explicitly in all cases. 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 $\mathrm{k}\mathrm{p}\mathrm{c}$ ($\mathrm{k}\mathrm{e}\mathrm{V}$/m)$(non-resonant),$\lambda_{fs}(0) \simeq 1.73 $\mathrm{k}\mathrm{p}\mathrm{c}$ ($\mathrm{k}\mathrm{e}\mathrm{V}$/m)$(resonant)\mathrm{.}IfWIMPswith$m_{wimp} \gtrsim 100 $\mathrm{G}\mathrm{e}\mathrm{V}$$decouplingat$T_d \gtrsim 10 $\mathrm{M}\mathrm{e}\mathrm{V}$$arepresentinthemixturewith$\nu_{wimp} \gg 10^{-12}$,$\lambda_{fs}(0) \lesssim 6.5 \times 10^{-3} $\mathrm{p}\mathrm{c}$$ 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.