Formation of trapped vacuum bubbles during inflation, and consequences for PBH scenarios

Abstract

A class of inflationary scenarios for primordial black hole (PBH) formation include a small barrier in the slope of the potential. There, the inflaton slows down, generating an enhancement of primordial perturbations. Moreover, the background solution overcomes the barrier at a very low speed, and large backward quantum fluctuations can prevent certain regions from overshooting the barrier. This leads to localized bubbles where the field remains trapped behind the barrier. In such models, therefore, we have two distinct channels for PBH production: the standard adiabatic density perturbation channel and the bubble channel. Here, we perform numerical simulations of bubble formation, addressing the issues of initial conditions, critical amplitude and bubble expansion. Further, we explore the scaling behaviour of the co-moving size of bubbles with the initial amplitude of the field fluctuation. We find that for small to moderate non-Gaussianity $f_{\rm NL}\lesssim 2.6$, the threshold for the formation of vacuum bubbles agrees with previous analytical estimates arXiv:1908.11357 to $5\%$ accuracy or so. We also show that the mass distribution for the two channels is different, leading to a slightly broader range of PBH masses. The bubble channel is subdominant for small $f_{\rm NL}$, and becomes dominant for $f_{\rm NL}\gtrsim 2.6$. We find that the mass of PBHs in the bubble channel is determined by an adiabatic overdensity surrounding the bubble at the end of inflation. Remarkably, the profile of this overdensity turns out to be of type-II. This represents a first clear example showing that overdensities of type-II can be dominant relative to the standard type-I. We also comment on exponential tails and on the fact that in models with local type non-Gaussianity (such as the one considered here), the occurrence of alternative channels can easily be inferred from unitarity considerations.

Figures (18)

• Figure 1: Left-panel: Inflationary potential $V(\phi)$ of Eq.\ref{['eq:pot_starobinsky']} in terms of $\phi/\phi_{\rm max}$. Right-panel: Shape of the inflationary potential around the bump for different values of $f_{\rm NL}$. The value $\phi_{\rm max}$ indicates the location of the local maxima of the bump and $V_{\rm max}= V(\phi_{\rm max})$. The parameters chosen can be found in Table \ref{['table:nuc']}.
• Figure 2: Schematic picture of a bubble formation. For fluctuations above the threshold $\mu_{b} > \mu_{b,c}$ a localized region will become trapped producing a vacuum bubble. In the opposite case, for sub-critical fluctuations $\mu_{b}< \mu_{b,c}$ the inflaton overshoots the barrier.
• Figure 3: Top panels: Dynamics of the inflaton field $\phi(N)$ (left) and $\dot{\phi}_{\rm bkg}(N)$ (right) for different values of $f_{\rm NL}$. The dashed vertical lines indicates the location of $N_{\rm max}$, $\phi_{\rm bkg}(N_{\rm max}) \equiv \phi_{\rm max}$. Bottom panels: Evolution of the Hubble slow-roll parameters $\epsilon_1$ (left), $\epsilon_{2}$ (right) respectively for different $f_{\rm NL}$ values. In all cases, the parameters taken correspond to $\nu_{b,c} \sim 8$ from Table \ref{['table:nuc']}.
• Figure 4: Left panel: Power spectrum of the Gaussian curvature fluctuation $\zeta_G$ for different values of $f_{\rm NL}$ for the case $\nu_{b,c} \sim 8$ (see Table \ref{['table:nuc']}). Right-panel: Reduction of the peak value of $\mathcal{P}_{\zeta_G}(k_{\rm max})$ in terms of $f_{\rm NL}$ for different values of $\nu_{b,c}$.
• Figure 5: Left panel: Evolution of the modes $\zeta_{G}$ in terms of the number of e-folds $N$ for some specific $k$ modes. The solid dots indicate the time when the modes cross the cosmological horizon at $k=a(N_{\rm crossing})H(N_{\rm crossing})$. We show $\epsilon_2$ as a dotted line. Notice that the modes that most contribute to the enhancement of $\mathcal{P}_{\zeta_G}$ become approximately frozen already at $N_{\star} \approx 41.18$, as given by the definition of Eq.\ref{['eq:Delta_N']}. This is in agreement with the expected behaviour once the attractor regime sets in. This is not exact as for instance the black dot is not completely frozen at the asymptotic value. The effect is more dramatic for the orange and pink dots but these are very subdominant in the power spectrum. Right-panel: Power spectrum of $\mathcal{P}_{\zeta_G}$ obtained solving the MS equation (blue line) together with the one obtained using the analytical approximation of Eq.\ref{['eq:slow_roll_equations']} (red line). The dots correspond to the values of $k$ shown in the left panel. In both cases, we have considered the case $f_{\rm NL} \approx 3.68$ with $\nu_{b,c} \sim 8$.