This paper presents an analysis of the predicted optical-to-X-ray spectral index (αox) within the context of ultra-luminous X-ray sources (ULXs) associated with stellar-mass black holes (BHs) and neutron stars (NSs). We used the population synthesis code COSMIC to simulate the evolution of binary systems and investigate the relationship between ultraviolet (UV) and X-ray emission during the ULX phase, namely the αox relation. Furthermore, we investigated the impact of metallicity on αox values. Notably, it predicts a significant anti-correlation between αox and UV luminosity (LUV), consistent with observations. The slope of this relationship varies with metallicity for black hole ULXs (BH-ULXs). The neutron star ULX (NS-ULX) population shows a relatively consistent slope around −0.33 across metallicities, with minor variations. The number of ULXs decreases with increasing metallicity, consistent with observational data. The X-ray luminosity function (XLF) shows a slight variation in its slope with metallicity, exhibiting a relative excess of high-luminosity ULXs at lower metallicities. The inclusion of the beaming effect in the analysis shows a significant impact on the XLF and αox, particularly at high accretion rates, where the emission is focused into narrower cones. We found that UV emission in ULXs is predominantly disc-dominated, which is the likely origin of the αox relation, with the percentage of disc-dominated ULXs increasing as metallicity rises.
Nyadzani et al. (Wed,) studied this question.