Orbital and super-orbital periods in ULX pulsars, disc-fed HMXBs, Be/X-ray binaries and double-periodic variables

We present evidence for a simple linear relationship between the orbital period and super-orbital period in ultra-luminous X-ray (ULX) pulsars, akin to what is seen in the population of disc-fed neutron star super-giant X-ray binary and Be/X-ray binary systems. We argue that the most likely cause of this relationship is the modulation of precessing hot spots or density waves in an accretion or circumstellar disc by the binary motion of the system, implying a physical link between ULX pulsars and high-mass X-ray binary (HMXB) pulsars. This hypothesis is supported by recent studies of Galactic and Magellanic Cloud HMXBs accreting at super-Eddington rates, and the position of ULX pulsars on the spin period--orbital period diagram of HMXBs. An interesting secondary relationship discovered in this work is the apparent connection between disc-fed HMXBs, ULXs and a seemingly unrelated group of early-type binaries showing so-called "double-periodic" variability. We suggest that these systems are good candidates to be the direct progenitors of Be/X-ray binaries.