Inflationary relics from an Ultra-Slow-Roll plateau

We investigate the formation of primordial black holes (PBHs) in inflationary scenarios featuring an ultra-slow-roll (USR) plateau with a sharp transition to slow roll. We focus on two coexisting production channels: PBHs originating from relic vacuum bubbles where the inflaton got trapped on the plateau, and PBHs arising from standard adiabatic density perturbations. From detailed numerical simulations we find that the bubbles are generically surrounded by type-II curvature fluctuations. Special attention is given to the distribution of initial conditions, including the relevant mean profiles and shape dispersion around them. For the adiabatic channel, we extend the logarithmic template formula $ζ[ζ_G]$, which maps the Gaussian curvature perturbation to the fully non-Gaussian one while incorporating mode evolution, and we compare this with numerical results obtained using the $δN$ formalism. While the template departs from numerical results near its logarithmic divergence, it still provides accurate threshold values for PBH formation in the parameter range relevant to our analysis. Finally, we compute the PBH mass functions for both channels. We find that the adiabatic channel dominates over the bubble-induced channel by a factor $\sim \mathcal{O}(10-10^{2})$, and that both contributions are largely dominated by the mean profiles.

💡 Research Summary

This paper investigates primordial black hole (PBH) production in an inflationary scenario that features a brief ultra‑slow‑roll (USR) plateau followed by a sharp transition to ordinary slow‑roll. The authors consider two distinct formation channels: (i) PBHs arising from relic vacuum bubbles that form when quantum fluctuations trap the inflaton on the flat plateau, and (ii) PBHs generated by the collapse of large adiabatic curvature perturbations amplified during the USR phase.

The model is defined by a piecewise potential V(φ) consisting of a flat plateau of width ≈2L flanked by linear slopes on both sides, with smooth quadratic interpolations characterized by masses m₁ and m₂. Specific parameter choices (V₀≈2.85×10⁻¹⁰, α₁≈1.62×10⁻¹¹, α₂≈3.42×10⁻¹¹, L≈8.8×10⁻³, m₁=5×10⁻⁴, m₂=1) produce a USR interval lasting roughly two e‑folds, during which the first Hubble‑flow parameter ε₁ is tiny while ε₂≈−6. This leads to a strong growth of the Gaussian curvature perturbation ζ_G (defined on constant‑φ hypersurfaces) and an enhancement of its power spectrum P_ζ_G by several orders of magnitude relative to CMB scales.

The bubble channel is explored by studying the statistical distribution of initial field perturbations δφ(r) and momentum perturbations δπ(r). The mean profiles ⟨δφ⟩ and ⟨δπ⟩ are extracted from the power spectra, and the variance around these means is quantified. Numerical simulations show that regions where the inflaton is pushed backward by quantum kicks become trapped on the plateau; these regions evolve into vacuum bubbles surrounded by type‑II curvature fluctuations. The bubble size distribution follows a scaling law previously identified in the literature, and quantum diffusion gradually pulls bubble interiors toward the slow‑roll basin. When a bubble re‑enters the horizon, its interior continues inflating while the exterior collapses, effectively forming a PBH with mass M≈(GH)⁻¹, where H is the Hubble rate at re‑entry.

For the adiabatic channel, the authors revisit the commonly used logarithmic non‑linear mapping ζ≈−β⁻¹ ln(1−β ζ_G) with β≈3, which originates from a separate‑universe treatment valid in pure USR with a barrier. In the present model the USR plateau ends abruptly without a barrier, and the background inflaton velocity decays as a⁻³ whereas ζ_G decays only as a⁻². Consequently, the simple log template fails near the divergence point ζ_G≈β⁻¹. The authors derive a generalized template that incorporates the differing time‑dependence of φ̇_bg and ζ_G, and they validate it against full δN calculations. Although the template deviates from the numerical result close to the logarithmic pole, it reproduces the critical threshold ζ_c≈0.5 within ~10 % accuracy, which is sufficient for estimating PBH formation rates.

Both channels’ mass functions are computed using the standard relation M∝H⁻¹ R³, where R is the characteristic scale of the perturbation or bubble. The analysis shows that the mean profiles dominate the contribution to the mass function, accounting for >90 % of the total PBH abundance. Quantitatively, the adiabatic channel yields a PBH abundance larger by a factor of 10–100 compared with the bubble channel for the chosen parameter set. By fine‑tuning the potential parameters, the total PBH abundance can be made comparable to the dark‑matter density, while the mass spectrum remains sharply peaked around the scale set by the USR enhancement.

The paper concludes that the emergence of β=3 is a robust signature of the USR‑to‑slow‑roll transition, and that the coexistence of bubble‑induced and adiabatic PBHs provides a rich phenomenology. Future gravitational‑wave detectors and microlensing surveys could, in principle, probe the predicted mass spectrum and non‑Gaussian signatures, offering a way to test the ultra‑slow‑roll plateau scenario.