The masses of active neutrinos in the $\nu$MSM from X-ray astronomy

In an extention of the Standard Model by three relatively light right-handed neutrinos (the $\nu$MSM model) the role of the dark matter particle is played by the lightest sterile neutrino. We demonstrate that the observations of the extragalactic X-ray background allow to put a strong upper bound on the mass of the lightest active neutrino and predict the absolute values of the mass of the two heavier active neutrinos in the $\nu$MSM, provided that the mass of the dark matter sterile neutrino is larger than $1.8$ keV.