Ultra-luminous X-ray sources and neutron-star-black-hole mergers from
  very massive close binaries at low metallicity
  Gravitational waves from the binary black hole (BH) merger GW150914 may
enlighten our understanding of ultra-luminous X-ray sources (ULXs), as
BHs>30Msun can reach luminosities>4x10^39 erg s^-1 without exceeding their
Eddington limit. It is then important to study variations of evolutionary
channels for merging BHs, which might instead form accreting BHs and become
ULXs. It was recently shown that massive binaries with mass ratios close to
unity and tight orbits can undergo efficient rotational mixing and evolve
chemically homogeneously, resulting in a compact BH binary. We study similar
systems by computing ~120000 detailed binary models with the MESA code covering
a wide range of initial parameters. For initial mass ratios M2/M1~0.1-0.4,
primaries >40Msun can evolve chemically homogeneously, remaining compact and
forming a BH without undergoing Roche-lobe overflow. The secondary then expands
and transfers mass to the BH, initiating a ULX phase. We predict that ~1 out of
10^4 massive stars evolves this way, and that in the local universe 0.13 ULXs
per Msun yr^-1 of star-formation rate are observable, with a strong preference
for low-metallicities. At metallicities log Z>-3, BH masses in ULXs are limited
to 60Msun due to the occurrence of pair-instability supernovae which leave no
remnant, resulting in an X-ray luminosity cut-off. At lower metallicities, very
massive stars can avoid exploding as pair-instability supernovae and instead
form BHs with masses above 130Msun, producing a gap in the ULX luminosity
distribution. After the ULX phase, neutron-star-BH binaries that merge in less
than a Hubble time are produced with a formation rate <0.2 Gpc^-3 yr^-1. We
expect that upcoming X-ray observatories will test these predictions, which
together with additional gravitational wave detections will provide strict
constraints on the origin of the most massive BHs that can be produced by
stars.
