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 nuMSM 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 nuMSM, provided that the mass of the dark matter sterile neutrino is larger than 1.8 keV.