Warm dark matter has, by definition, a velocity dispersion. Let vhms(a)=vhms(1)/a be the root-mean-square velocity of non-relativistic warm dark matter particles in the early universe at expansion parameter a. vhms(1) is an adiabatic invariant. We obtain vhms(1)？in the core of 11 dwarf galaxies dominated by dark matter, from their observed rotation curves, up to a rotation and relaxation correction. We obtain a mean 0.490 km/s and standard deviation 0.160 km/s, with a distribution peaked at the lower end. We apply a mild, data driven, rotation and relaxation correction that obtains the adiabatic invariant in the core of the galaxies: vhms(1)=0.406 ± 0.069 km/s. These two small relative standard deviations justify the prediction that the adiabatic invariant vhms(1) in the core of the galaxies is of cosmological origin if dark matter is warm. This result is in agreement with measurements of vhms(1) based on spiral galaxy rotation curves, galaxy ultra-violet luminosity distributions, galaxy stellar mass distributions, the formation of first galaxies, reionization, and the velocity dispersion cut-off mass.
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