Gravitational-wave signatures of primordial black hole clusters and the imprint of an early dense-core collapse

Abstract: Primordial black holes (PBHs) are expected to be born essentially non-spinning and, if the primordial curvature fluctuations possess non-Gaussian tails, strongly clustered. Any spin measured in a PBH population must therefore be dynamically generated. Using direct $N$-body simulations with \textsc{nbody6++gpu}, modified to treat nearly parabolic dynamical-capture mergers, we quantify the conditions under which an initially non-spinning PBH cluster develops the spin, mass and stochastic-background signatures that gravitational-wave observations are now sensitive to. We find a sharp tension: producing a root-mean-square spin $χ<em>{\rm rms}\sim0.1 - 0.2$ in the $30 - 50\,M</em>\odot$ range requires initial Plummer radii of order $10^{-5}\,{\rm pc}$, whereas survival over cosmological times requires parsec-scale clusters. This tension is resolved naturally if PBHs form with a compact, short-lived core embedded in an extended halo, and we therefore simulate the two regimes as independent blocks. The core block ($4000$ PBHs, Plummer radii $$3\times10^{-6} - 10^{-4}\,{\rm pc}$$) collapses in days to years, producing a burst of mergers whose orbits are strongly non-circular ($e\to1$) and of low dimensionless orbital angular momentum, so that remnant spins are systematically below the quasi-circular value $χ\simeq0.7$. Redshifted from their formation epoch, the core mergers deposit a stochastic background peaking in the $10^{{-6}$--$10^{-4}\,{\rm} Hz}$ band accessible to LISA. The picture predicts a specific division of labour between observables, spins set in the primordial collapse, detectable mergers set by the long-term halo, and a two-component stochastic background, and, in particular, that both the spin and the mass data favour a primordial mass function truncated near $50\,M_\odot$.