Limits on hierarchical black hole mergers from the most negative ${\chi }_{\mathrm{eff}}$ systems

It has been proposed that some black holes (BHs) in binary black hole (BBH) systems are born from ``hierarchical mergers" (HM); i.e. earlier mergers of smaller BHs. These HM products have spin magnitudes $\chi \sim 0.7$, and, if they are dynamically assembled into BBH systems, their spin orientations will be sometimes anti-aligned with the binary orbital angular momentum. In fact, as Baibhav et al. (2020) showed, $\sim 16\mathrm{\%}$ of BBH systems that include HM products will have an effective inspiral spin parameter, ${\chi }_{\mathrm{eff}}<-0.3$. Nevertheless, the LIGO-Virgo-Kagra (LVK) gravitational-wave (GW) detectors have yet to observe a BBH system with ${\chi }_{\mathrm{eff}}\lesssim -0.2$, leading to upper limits on the fraction of HM products in the population. We fit the astrophysical mass and spin distribution of BBH systems and measure the fraction of BBH systems with ${\chi }_{\mathrm{eff}}<-0.3$, which implies an upper limit on the HM fraction. We find that fewer than $26\mathrm{\%}$ of systems in the underlying BBH population include HM products (90\%. credibility). Even among BBH systems with primary masses $m_1=60M_{\odot }$, the HM fraction is less than 69\%, which may constrain the location of the pair-instability mass gap. With 300 GW events (to be expected in the LVK's next observing run), if we fail to observe a BBH with ${\chi }_{\mathrm{eff}}<-0.3$, we can conclude that the HM fraction is smaller than ${2.5}_{-2.2}^{+9.1}\mathrm{\%}$.