How Cold is Cold Dark Matter?

If cold dark matter consists of particles, these must be non-interacting and non-relativistic by definition. In most cold dark matter models however, dark matter particles inherit a non-vanishing velocity dispersion from interactions in the early universe, a velocity that redshifts with cosmic expansion but certainly remains non-zero. In this article, we place model-independent constraints on the dark matter temperature to mass ratio, whose square root determines the dark matter velocity dispersion. We only assume that dark matter particles decoupled kinetically while non-relativistic, when galactic scales had not entered the horizon yet, and that their momentum distribution has been Maxwellian since that time. Under these assumptions, using cosmic microwave background and matter power spectrum observations, we place upper limits on the temperature to mass ratio of cold dark matter today (away from collapsed structures). These limits imply that the present cold dark matter velocity dispersion has to be smaller than 54 m/s. Cold dark matter has to be quite cold, indeed.