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

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