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.

Preprint

Comment: 29 pages + 12 figures

Subjects: Astrophysics - Cosmology and Nongalactic Astrophysics; General Relativity and Quantum Cosmology; High Energy Physics - Phenomenology