Context : Ultra-luminous X-ray sources ( ULXs ) are X-ray sources located off the galactic centre and have luminosities exceeding the Eddington limit of a stellar-mass black hole ( L _ { X } > 10 ^ { 39 } { erg s } ^ { -1 } ) .
Observed X-ray variability suggests that ULXs are X-ray binary systems .
With the discovery of X-ray pulsations in some of these objects ( e.g .
M82 X-2 ) , a certain fraction of the ULX population is implied to have a neutron star as the accretor .

Aims : We present systematic modelling of intermediate-mass X-ray binaries ( IMXBs ; donor-star mass range 2.0 – 8.0 M _ { \odot } and neutron star accretors ) to explain the formation of this sub-population of ULXs .

Methods : Using MESA , we explore the allowed initial parameter space of binary systems consisting of a neutron star and an intermediate-mass donor star , that could explain the observed properties of ULXs .
These donors are transferring mass at super-Eddington rates while the accretion is limited locally in the accretion disk by the Eddington limit .
Thus , our simulations take into account beaming effects and also include stellar rotation , tides , general angular momentum losses , and a detailed and self-consistent calculation of the mass-transfer rate .

Results : Exploring the initial parameters that lead to the formation of neutron-star ULXs , we study the conditions that lead to dynamical stability of these systems , which depends strongly on the response of the donor star to mass loss .
Using two values for the initial neutron star mass ( 1.3 M _ { \odot } and 2.0 M _ { \odot } ) we present two sets of mass-transfer calculation grids for comparison with observations of NS ULXs .
We find that IMXBs can produce NS-ULXs with typical time-averaged isotropic-equivalent X-ray luminosities between 10 ^ { 39 } – 10 ^ { 41 } { erg s } ^ { -1 } on a timescale of up to \sim 1.0 { Myr } for the lower luminosities .
Finally , we estimate their likelihood of detection , the types of white-dwarf remnants left behind by the donors , as well as the total amount of mass accreted by the neutron stars .

Conclusions : We show that observed super-Eddington luminosities can be achieved in IMXBs undergoing non-conservative mass transfer , assuming geometrical beaming .
We also compare our results to the observed pulsating ULXs and infer their initial parameters .
Our results suggest that a large subset of the observed pulsating ULX population can be explained by IMXBs in a super-Eddington mass-transfer phase .