Chandra observations of "the antennae" galaxies (NGC 4038/4039).: IV.: The X-ray source luminosity function and the nature of ultraluminous X-ray sources

We derive the X-ray luminosity function (XLF) of the X-ray source population detected in the Chandra observation of NGC 4038/4039 (the Antennae). We explicitly include photon-counting and spectral parameter uncertainties in our calculations. The cumulative XLF is well represented by a at power law (alpha = -0.47), similar to those describing the XLFs of other star-forming systems (e.g., M82, the disk of M81) but different from those of early-type galaxies. This result associates the X-ray source population in the Antennae with young high-mass X-ray binaries. In comparison with less actively star-forming galaxies, the XLF of the Antennae has a highly significant excess of sources with luminosities above 10(39) ergs s(-1) (ultraluminous sources; ULXs). We discuss the nature of these sources, based on the XLF and on their general spectral properties, as well as their optical counterparts discussed in Paper III. We conclude that the majority of the ULXs cannot be intermediate-mass black hole (M > 10 1000 M.) binaries, unless they are linked to the remnants of massive Population III stars (the Madau & Rees model). Instead, their spatial and multiwave-length properties can be well explained by beamed emission as a consequence of supercritical accretion. Binaries with a neutron star or moderate-mass black hole (up to 20 M.) and B2 to A type star companions would be consistent with our data. In the beaming scenario, the XLF should exhibit characteristic breaks that would be visible in future deeper observations of the Antennae.