Formation of primordial black holes from non-Gaussian perturbations produced in a waterfall transition

TL;DR

The paper investigates primordial black hole formation from strongly non-Gaussian curvature perturbations generated during a waterfall transition at the end of hybrid inflation. It derives a non-Gaussian ζ distribution, ζ = -A (χ^2 - ⟨χ^2⟩), and applies a Press-Schechter framework to compute PBH abundances and mass spectra as functions of the inflationary parameters, notably β ≡ |m_χ^2|/H^2. The results show a strong sensitivity of PBH production to the peak amplitude and the formation threshold ζ_c, yielding lower bounds on β (around $β \gtrsim 1.6$–$2.3$) and indicating that only with fine-tuning can heavier PBHs be produced; the model can naturally yield light PBHs ($M_{BH} \sim 10^5$ g) with $β_{PBH} \sim 10^{-3}$, potentially contributing to a high-frequency gravitational wave background. This work provides a tractable framework to constrain hybrid inflation parameters using PBH limits and informs the prospects for PBH-related gravitational-wave signatures.

Abstract

We consider the process of primordial black hole (PBH) formation originated from primordial curvature perturbations produced during waterfall transition (with tachyonic instability), at the end of hybrid inflation. It is known that in such inflation models, rather large values of curvature perturbation amplitudes can be reached, which can potentially cause a significant PBH production in the early Universe. The probability distributions of density perturbation amplitudes in this case can be strongly non-Gaussian, which requires a special treatment. We calculated PBH abundances and PBH mass spectra for the model, and analyzed their dependence on model parameters. We obtained the constraints on the parameters of the inflationary potential, using the available limits on $β_{PBH}$.

Figures (4)

• Figure 1: The probability distribution $p_{\zeta}(\zeta)$ for $\Sigma=0.7$, ${\cal P}_\zeta^0=0.4$ and for different values of $R$: from left to right, $R=10 k_0^{-1}, k_0^{-1}, 0.1 k_0^{-1}$. The possible (model dependent) values of $\zeta_c$ ($\zeta_c=0.75$ and $\zeta_c=1$) are shown by dashed lines.
• Figure 2: The probability $P(\zeta > \zeta_c)$ as a function of $R$. From top to bottom, ${\cal P}_\zeta^0 = 0.4 (\zeta_{max}\approx 0.9), 0.28 (\zeta_{max}\approx 0.752), 0.27846 (\zeta_{max}\approx 0.750012)$. For all curves, $\Sigma=0.7$, $\zeta_c=0.75$.
• Figure 3: The PBH mass spectra for different values of the perturbation spectrum amplitudes. From right to left, ${\cal P}_\zeta^0=1 (\zeta_{max}\approx 1.42), 0.4 (\zeta_{max}\approx 0.9), 0.28 (\zeta_{max}\approx 0.752), 0.27846 (\zeta_{max}\approx 0.750012), 0.2784511 (\zeta_{max}\approx 0.750000068)$. The position of the peak in ${\cal P}_\zeta(k)$-spectrum is the same for all cases. For the calculation we used the value $\Sigma=0.7$, and $\zeta_c=0.75$. The mass $M_h^0$ corresponds to horizon mass at the moment of time when perturbation with comoving wave number $k_0$ enters horizon.
• Figure 4: The horizon mass regions corresponding to the position of the peak in curvature perturbation power spectrum (shaded areas). a)$\; \beta\approx 2.3, \zeta_c=0.75$; b)$\beta\approx 1.65, \zeta_c=1$. In both cases, the value of $\beta$ is just enough to produce PBHs, and we vary the parameter $\phi_c$ (or, equivalently, $\gamma$) between minimal and maximal possible values. The exact PBH abundance and relation of characteristic PBH mass $M_{BH}$ to $M_h$ will depend on values of parameters in a fine-tuning regime (see Fig. \ref{['fig-nBH']} for an illustration).