Very massive stars, pair-instability supernovae and intermediate-mass black holes with the SEVN code

Understanding the link between massive ($\gtrsim 30$ M$_{\odot{}}$) stellar black holes (BHs) and their progenitor stars is a crucial step to interpret observations of gravitational-wave events. In this paper, we discuss the final fate of very massive stars (VMSs), with zero-age main sequence (ZAMS) mass $>150$ M$_{\odot}$, accounting for pulsational pair-instability supernovae (PPISNe) and for pair-instability supernovae (PISNe). We describe an updated version of our population synthesis code SEVN, in which we added stellar evolution tracks for VMSs with ZAMS mass up to $350$M$_{\odot{}}$ and we included analytical prescriptions for PPISNe and PISNe. We use the new version of SEVN to study the BH mass spectrum at different metallicity $Z$, ranging from $Z=2.0\times 10^{-4}$ to $Z=2.0\times 10^{-2}$. The main effect of PPISNe and PISNe is to favour the formation of BHs in the mass range of the first gravitational-wave event (GW150914), while they prevent the formation of remnants with mass 60 - 120 M$_{\odot{}}$. In particular, we find that PPISNe significantly enhance mass loss of metal-poor ($Z\leq 2.0\times 10^{-3}$) stars with ZAMS mass $60\leq$M$_{\mathrm{ZAMS}}/$M$_{\odot{}}\leq 125$. In contrast, PISNe become effective only for moderately metal-poor ($Z<8.0\times 10^{-3}$) VMSs. VMSs with M$_{\rm ZAMS}\gtrsim{}220$ M$_\odot$ and $Z<10^{-3}$ do not undergo PISNe and form intermediate-mass BHs (IMBHs, with mass $\gtrsim 200$M$_{\odot{}}$) via direct collapse.