Initial clustering of primordial black holes: A general formulation for arbitrary local non-Gaussianity

Initial spatial clustering of primordial black holes (PBHs) induced by local-type non-Gaussianity (LNG) can substantially modify cosmological constraints on PBH abundance. Existing formulations of initial PBH clustering are primarily restricted to either small non-Gaussianity or specific regimes, and a unified theoretical framework has been lacking. Here, we present a general analytical formulation for the PBH two-point correlation function applicable to arbitrary LNGs. Under the assumptions that PBHs form only at the large peaks of perturbations, and that long-wavelength modes weakly modulate the local variance of short-wavelength (PBH-scale) perturbations, we derive an analytic expression for the PBH bias parameter, directly connecting initial clustering to the primordial trispectrum in the collapsed limit. We demonstrate the versatility of our formula by computing the bias parameters in the ultra-slow-roll inflation, curvaton, and modulated reheating scenarios. We also formally generalize the framework to broad power spectra to account for correlations across different PBH mass scales. Because our formulation does not rely on weak or perturbative non-Gaussianity assumptions, our result provides a universal theoretical basis for evaluating initial clustering impact on PBH observables.