  Weakly interacting massive particles (WIMPs) are arguably the most natural DM
candidates from a particle physics point of view. After their number density
has frozen out in the early universe, determining their relic density today,
WIMPs are still kept very close to thermal equilibrium by scattering events
with standard model particles. The complete decoupling from the thermal bath
happens as late as around 1-10 MeV, thereby setting an important cosmological
scale that can directly be translated into a small-scale cutoff of the spectrum
of matter density fluctuations. We present here a full analytic treatment of
the kinetic decoupling process from first principles. This allows an exact
determination of the decoupling scale, for an arbitrary WIMP candidate and any
scattering amplitude. As an application, we consider the situation of the
lightest supersymmetric particle as well as the lightest Kaluza-Klein particle
that arises in theories with universal extra dimensions; furthermore, we show
that our formalism can also easily be applied to, e.g., the evolution of the
non-relativistic electrons into the recombination regime. Finally, we comment
on the impacts for the smallest gravitationally bound structures and the
prospects for the indirect detection of dark matter.
