Stochastic-tail of the curvature perturbation in hybrid inflation

TL;DR

The paper investigates the tail of the primordial curvature perturbation PDF in mild-waterfall hybrid inflation using the stochastic-$\delta\mathcal{N}$ formalism, confirming exponential tails and uncovering an effective upper bound on $\delta\mathcal{N}$ tied to exact hilltop trajectories. It introduces Johnson's $S_U$-distribution as a flexible fit for the PDF and demonstrates that balancing the inflaton potential (including cubic terms) can substantially suppress perturbations, sometimes below the PBH formation threshold. The results show that in balanced quadratic models the exponential tail remains but PBH production is strongly constrained, while Cubic terms can lead to dramatic suppression and require non-perturbative treatment for accurate predictions. These findings have important implications for PBH abundance estimates, peak-theory applications, and potentially halo formation, and they highlight the necessity of non-perturbative stochastic methods in multi-field inflation analyses.

Abstract

The exponential-tail behaviours of the probability density function (PDF) of the primordial curvature perturbation are confirmed in the mild-waterfall variants of hybrid inflation with the use of the stochastic formalism of inflation. On top of these tails, effective upper bounds on the curvature perturbation are also observed, corresponding to the exact hilltop trajectory during the waterfall phase. We find that in the model where the leading and higher-order terms in the expansion of the inflaton potential around the critical point are fine-tuned to balance, this upper bound can be significantly reduced, even smaller than the primordial black hole (PBH) threshold, as a novel perturbation-reduction mechanism than the one proposed by Tada and Yamada. It makes PBH formation much difficult compared to the Gaussian or exponential-tail approximation. We also introduce Johnson's $S_U$-distribution as a useful fitting function for the PDF, which reveals a nonlinear mapping between the Gaussian field and the curvature perturbation.

Figures (1)

• Figure 1: The PDF of the curvature perturbation $\zeta=\delta\mathcal{N}=\mathcal{N}-\expval{\mathcal{N}}$ for "Quadratic $n=2$" (top-left), "Quadratic $n=15$" (top-right), and "Cubic $n=2$" (bottom) with the model parameters listed in Table \ref{['tab: model parameters']}. Blue dots are numerical results with $10^7$ samples. Error bars are estimated by the jackknife resampling. That is, the $10^7$ samples are divided into ten data sets of $10^6$ samples. The PDF is computed for each data set and the error is estimated by the standard error for those ten PDF data. Orange dashed lines represent Johnson's $S_U$-distribution fitting, whose fitting parameters are listed in Table \ref{['tab: Johnson parameters']}. The vertical thin lines at $\delta\mathcal{N}=2.11$ in the top-left panel and at $\delta\mathcal{N}=0.20$ in the bottom panel correspond to the exact hilltop trajectories $\psi_\mathrm{r}=0$, which represent effective upper bounds of $\delta\mathcal{N}$. In fact, the PDF value significantly drops beyond these bounds, and the small enough error bars ensure the robustness of these behaviours. The $n=15$ case also has a similar upper bound but at $\delta\mathcal{N}=3.02$ and hence out of the plot range.