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

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 $\text{textsc}nbody6++gpu$, 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 ${\chi }_{\mathrm{r}\mathrm{m}\mathrm{s}}\sim 0.1-0.2$ in the $30-50M_{\odot }$ range requires initial Plummer radii of order ${10}^{-5}\mathrm{p}\mathrm{c}$, 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 $3\times {10}^{-6}-{10}^{-4}\mathrm{p}\mathrm{c}$) collapses in days to years, producing a burst of mergers whose orbits are strongly non-circular ($e\to 1$) and of low dimensionless orbital angular momentum, so that remnant spins are systematically below the quasi-circular value $\chi \simeq 0.7$. Redshifted from their formation epoch, the core mergers deposit a stochastic background peaking in the ${10}^{-6}$–${10}^{-4}\mathrm{H}\mathrm{z}$ 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 $50M_{\odot }$.