Exact PBH abundance for the full axion cosine potential

Derive the primordial-black-hole abundance and mass spectrum for the full axion cosine potential in the non-perturbative regime of large initial misalignment angle, where the harmonic approximation fails and the curvature perturbation becomes strongly non-linear.

Background

The analysis computes primordial-black-hole formation using the quadratic approximation to the axion potential and restricts the initial misalignment angle to approximately θ≲0.3\theta\lesssim 0.3. For larger angles, higher-order terms in the cosine potential are unsuppressed, producing strongly non-linear curvature perturbations that are not reliably described by the employed Gaussian-based collapse-threshold treatment.

The unresolved task is to perform an exact treatment of the full cosine potential and determine the resulting primordial-black-hole abundance when the axion dynamics is intrinsically non-perturbative.

References

Consequently, throughout this work, we restrict our analysis to the perturbative regime (as reflected in our chosen parameter sets where \theta \lesssim 0.3) where the quadratic approximation remains safely valid, deferring the exact treatment of the full cosine potential to future studies (see Appendix \ref{app:PBH} for further discussion).

— Primordial Black Hole and Gravitational Wave by Peaked Cosmic Perturbations from Axion Curvaton  (2610.00457 - Harigaya et al., 30 Sep 2026) in Section 3.1, subsection “The NG PBH mass distribution at all orders”