Signatures of Hierarchical Mergers in Black Hole Spin and Mass distribution

Recent gravitational wave (GW) observations by LIGO/Virgo show evidence for hierarchical mergers, where the merging BHs are the remnants of previous BH merger events. These events may carry important clues about the astrophysical host environments of the GW sources. In this paper, we present the distributions of the effective spin parameter ($\chi_\mathrm{eff}$), the precession spin parameter ($\chi_\mathrm{p}$), and the chirp mass ($m_\mathrm{chirp}$) expected in hierarchical mergers. Under a wide range of assumptions, hierarchical mergers produce (i) a monotonic increase of the average of the typical total spin for merging binaries, which we characterize with ${\bar \chi}_\mathrm{typ}\equiv \overline{(\chi_\mathrm{eff}^2+\chi_\mathrm{p}^2)^{1/2}}$, up to roughly the maximum $m_\mathrm{chirp}$ among first-generation (1g) BHs, and (ii) a plateau at ${\bar \chi}_\mathrm{typ}\sim 0.6$ at higher $m_\mathrm{chirp}$. We suggest that the maximum mass and typical spin magnitudes for 1g BHs can be estimated from ${\bar \chi}_\mathrm{typ}$ as a function of $m_\mathrm{chirp}$. The GW data observed in LIGO/Virgo O1--O3a prefers an increase in ${\bar \chi}_\mathrm{typ}$ at low $m_\mathrm{chirp}$, which is consistent with the growth of the BH spin magnitude by hierarchical mergers, at $\sim 2 \sigma$ confidence. A Bayesian analysis suggests that 1g BHs have the maximum mass of $\sim 15$--$30\,M_\odot$ if the majority of mergers are of high-generation BHs (not among 1g-1g BHs), which is consistent with mergers in active galactic nucleus disks and/or nuclear star clusters, while if mergers mainly originate from globular clusters, 1g BHs are favored to have non-zero spin magnitudes of $\sim 0.3$. We also forecast that signatures for hierarchical mergers in the ${\bar \chi}_\mathrm{typ}$ distribution can be confidently recovered once the number of GW events increases to $\gtrsim O(100)$.