Inflationary relics from an Ultra-Slow-Roll plateau

TL;DR

The paper investigates primordial black hole formation in a single-field inflation model featuring a ultra-slow-roll plateau, focusing on two coexisting channels: relic vacuum bubbles and adiabatic density perturbations.It develops a statistical and numerical framework to characterize initial-condition shapes, their dispersion, and the non-linear evolution that leads to PBH formation, including a generalized $\zeta[\zeta_G]$ template for USR dynamics.The results show that adiabatic perturbations dominate the PBH abundance by about one to two orders of magnitude, while bubble-induced PBHs are subdominant but robust and tied to Type-II fluctuations; diffusion processes create a fractal sponge that preserves the original PBH scale.The study provides mass-function predictions and suggests gravitational waves as cross-checks, with avenues for generalization to broader plateau shapes and stochastic effects.

Abstract

We investigate the formation of primordial black holes (PBHs) in inflationary scenarios featuring an ultra-slow-roll (USR) plateau, focusing 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.

Figures (11)

• Figure 1: Inflationary potential $V(\phi)$ as a function of $\phi$. The cyan circles represent inflationary trajectories, which can undergo a backward quantum fluctuation (red arrow) and become trapped, forming a vacuum relic, instead of rolling down from the flat plateau (blue arrow).
• Figure 2: Left-panel: Field $\phi_{\rm bkg}(N)$. Middle panel: Veolocity field $\dot{\phi}_{\rm bkg}(N)$. Right panel: $\epsilon_2(N)$ parametter.
• Figure 3: Power spectra of the different previously defined quantities in terms of $k/k_*$, where $k_{*} = a(N_*) H(N_*)$. The spectra are evaluated at $N_*$, except for the blue curve, which is evaluated at the end of inflation, $N_{\rm end}$.
• Figure 4: Mode evolution of $\delta \phi$ and $\delta \pi$, showing their real and imaginary components as a function of the number of e-folds. The three vertical dashed lines correspond to the times $N_k$ when the wave mode $k$ re-enters the horizon, for the case of $N_{*}$ (black), the beginning of the USR phase $N_{\rm USR}$ (red line), and the location of the maximum of the power spectrum $\mathcal{P}_{\zeta_G}(k)$ at $k_{\rm peak}$ (blue). Left panel corresponds to $k=k_{\rm peak}$ and right panel $k=k_{*}$.
• Figure 5: Top-Left panel: Normalized correlators $\Psi_{\phi},\Psi_{\pi}$. Top-Right panel: Ratio between the correlators. Bottom-Left panel: Dispersion shapes $\Delta_{\phi}(r), \Delta_{\pi}(r)$. Bottom-Right panel: Shapes $\delta \phi(r,n,m),\delta \pi(r,n,m)$ taking $\mu \approx 2.613 \cdot 10^{-5}$. We have also used $\sigma_{\delta \phi}\approx 2.836 \cdot 10^{-6}$, $\tilde{\sigma}_{\pi} \approx 4.960 \cdot 10^{-7}$