Pulsing ULXs: tip of the iceberg?

We consider the three currently knownpulsing ultraluminous X-ray sources (PULXs). We show that in one of them the observed spin-up rate requires super-Eddington accretion rates at the magnetospheric radius, even if magnetar-strength fields are assumed. In the two other systems, a normal-strength neutron star field implies super-Eddington accretion at the magnetosphere. Adopting super-Eddington mass transfer as the defining characteristic of ULX systems, we find the parameters required for self-consistent simultaneous fits of the luminosities and spin-up rates of the three pulsed systems. These imply near equality between their magnetospheric radii R-M and the spherization radii R-sph where radiation pressure becomes important and drives mass-loss from the accretion disc. We interpret this near equality as a necessary condition for the systems to appear as pulsed, since if it is violated the pulse fraction is small. We show that as a consequence all PULXs must have spin-up rates nu greater than or similar to 10(-10) s(-2), an order of magnitude higher than in any other pulsing neutron-star binaries. The fairly tight conditions required for ULXs to show pulsing support our earlier suggestion that many unpulsed ULX systems must actually contain neutron stars rather than black holes.