Constraining hierarchical mergers of binary black holes detectable with LIGO-Virgo

Most of the binary black hole (BBH) mergers detected by LIGO and Virgo could be explained by first-generation mergers formed from the collapse of stars, while others might come from second (or higher) generation mergers, namely hierarchical mergers, with at least one of the black holes (BHs) being the remnant of a previous merger. A primary condition for the occurrence of hierarchical mergers is that the remnant BHs can be retained by the host star cluster. We present a simple formula to estimate the hierarchical merger rate in star clusters. We find this latter to be proportional to the retention probability. Further, we show that $\sim 2\%$ of BBH mergers in nuclear star clusters (NSCs) may instead be of hierarchical mergers, while the percentage in globular clusters (GCs) is only a few tenths of a percent. However, the rates of hierarchical merger in GCs and NSCs are about the same, namely of $\sim {\mathcal O(10^{-2})}~{\rm Gpc^{-3}~yr^{-1}}$, because the total BBH merger rate in GCs is larger than that in NSCs. This suggests that if a gravitational-wave event detected by LIGO-Virgo is identified as a hierarchical merger, then it is equally likely that this merger originated from a GC or an NSC.