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Gravitational waves from the binary black hole (BH) merger GW150914 mayenlighten our understanding of ultra-luminous X-ray sources (ULXs), asBHs>30Msun can reach luminosities>4x10^39 erg s^-1 without exceeding theirEddington limit. It is then important to study variations of evolutionarychannels for merging BHs, which might instead form accreting BHs and becomeULXs. It was recently shown that massive binaries with mass ratios close tounity and tight orbits can undergo efficient rotational mixing and evolvechemically homogeneously, resulting in a compact BH binary. We study similarsystems by computing ~120000 detailed binary models with the MESA code coveringa wide range of initial parameters. For initial mass ratios M2/M1~0.1-0.4,primaries >40Msun can evolve chemically homogeneously, remaining compact andforming a BH without undergoing Roche-lobe overflow. The secondary then expandsand transfers mass to the BH, initiating a ULX phase. We predict that ~1 out of10^4 massive stars evolves this way, and that in the local universe 0.13 ULXsper Msun yr^-1 of star-formation rate are observable, with a strong preferencefor low-metallicities. At metallicities log Z>-3, BH masses in ULXs are limitedto 60Msun due to the occurrence of pair-instability supernovae which leave noremnant, resulting in an X-ray luminosity cut-off. At lower metallicities, verymassive stars can avoid exploding as pair-instability supernovae and insteadform BHs with masses above 130Msun, producing a gap in the ULX luminositydistribution. After the ULX phase, neutron-star-BH binaries that merge in lessthan a Hubble time are produced with a formation rate <0.2 Gpc^-3 yr^-1. Weexpect that upcoming X-ray observatories will test these predictions, whichtogether with additional gravitational wave detections will provide strictconstraints on the origin of the most massive BHs that can be produced bystars.

Ultra-luminous X-ray sources and neutron-star-black-hole mergers from very massive close binaries at low metallicity