## Free streaming of particles

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To quantify the clustering properties of DM one defines a comoving free streaming horizon$\lambda _{FS}$ via
where $a(t)$ is a scale factor, $t_0$ is the time today (see the age of the Universe) and $v(t)$ is the velocity of DM particles. The velocity is given by the following expression
where $c$ is the speed of light and time of non-relativistic transition$t_{nr}$ is defined as the time when
where $m$ is the mass of the DM particle.