How Beaming Shapes the Demographics of Ultraluminous X-ray Sources?

Ultraluminous X-ray sources (ULXs) are off-nuclear compact objects with apparent luminosities above 10^39 erg/s, often exceeding the Eddington limit for stellar-mass black holes. Beaming is a commonly invoked mechanism to explain their extreme brightness, and the dependence of the beaming factor on accretion rate is a critical parameter. In this work, we investigate how different beaming prescriptions affect the predicted properties of ULX populations. Using binary population synthesis, we construct synthetic X-ray luminosity functions (XLFs) for both classical and log-modified beaming models at solar and sub-solar metallicities. The classical model predicts a larger intrinsic number of bright ULXs, but strong beaming reduces their observable fraction, resulting in fewer visible ULXs compared to the log-modified model. The log-modified prescription yields a shallower slope at high-luminosity, aligning better with observed XLFs, and increases the fraction of observable neutron star ULXs above 10^39 erg/s. These results underscore the significant role of the beaming law in shaping ULX statistical distributions and assessing neutron star contributions to the population.