Primordial non-Gaussianity up to all orders. Theoretical aspects and implications for primordial black hole models

We develop an exact formalism for the computation of the abundance of primordial black holes (PBHs) in the presence of local non-gaussianity (NG) in the curvature perturbation field. For the first time, we include NG going beyond the widely used quadratic and cubic approximations, and consider a completely generic functional form. Adopting threshold statistics of the compaction function, we address the computation of the abundance both for narrow and broad power spectra. While our formulas are generic, we discuss explicit examples of phenomenological relevance considering the physics case of the curvaton field. We carefully assess under which conditions the conventional perturbative approach can be trusted. In the case of a narrow power spectrum, this happens only if the perturbative expansion is pushed beyond the quadratic order (with the optimal order of truncation that depends on the width of the spectrum). Most importantly, we demonstrate that the perturbative approach is intrinsically flawed when considering broad spectra, in which case only the non-perturbative computation captures the correct result. Finally, we describe the phenomenological relevance of our results for the connection between the abundance of PBHs and the stochastic gravitational wave (GW) background related to their formation. As NGs modify the amplitude of perturbations necessary to produce a given PBHs abundance and boost PBHs production at large scales for broad spectra, modelling these effects is crucial to connect the PBH scenario to its signatures at current and future GWs experiments.