X-ray binaries with neutron stars at different accretion stages

Different accretion regimes onto magnetized NSs in HMXBs are considered: wind-fed supersonic (Bondi) regime at high accretion rates $\dot M\gtrsim 4\times 10^{16}$~[g s$^{-1}$] , subsonic settling regime at lower $\dot M$ and supercritical disc accretion during Roche lobe overflow. In wind-fed stage, NSs in HMXBs reach equilibrium spin periods $P^*$ proportional to binary orbital period $P_b$. At supercritical accretion stage, the system may appear as a pulsating ULX. Population synthesis of Galactic HMXBs using standard assumptions on the binary evolution and NS formation is presented. Comparison of the model $P^*$ -- $P_b$ (the Corbet diagram), $P^*$ -- $L_x$ and $P_b$ -- $L_x$ distributions with those for the observed HMXBs (including Be X-ray binaries) and pulsating ULXs suggests the importance of the reduction of $P^*$ in non-circular orbits, explaining the location of Be X-ray binaries in the model Corbet diagram, and the universal parameters of pulsating ULXs depending only on the NS magnetic fields.