Weakly interacting massive particles are part of the lepton-photon plasma in the early universe until kinetic decoupling , after which time the particles behave like a collisionless gas with nonzero temperature .
The Boltzmann equation for WIMP-lepton collisions is reduced to a Fokker-Planck equation for the evolution of the WIMP distribution including scalar density perturbations .
This equation and the Einstein and fluid equations for the plasma are solved numerically including the acoustic oscillations of the plasma before and during kinetic decoupling , the frictional damping occurring during kinetic decoupling , and the free-streaming damping occurring afterwards and throughout the radiation-dominated era .
An excellent approximation reduces the solution to quadratures for the cold dark matter density and velocity perturbations .
The subsequent evolution is followed through electron pair annihilation and the radiation-matter transition ; analytic solutions are provided for both large and small scales .
For a 100 GeV WIMP with bino-type interactions , kinetic decoupling occurs at a temperature T _ { d } = 23 MeV .
The transfer function in the matter-dominated era leads to an abundance of small cold dark matter halos ; with a smooth window function the Press-Schechter mass distribution is dn / d \ln M \propto M ^ { -1 / 3 } for M < 10 ^ { -4 } ( T _ { d } / 10 MeV ) ^ { -3 } M _ { \odot } .