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Threshold for PBH formation in the type-II region and its analytical estimation

Albert Escrivà

Abstract

We numerically simulate the formation of Primordial Black Holes (PBHs) in a radiation-dominated Universe under the assumption of spherical symmetry, driven by the collapse of adiabatic fluctuations, for different curvature profiles $ζ$. Our results show that the threshold for PBH formation, defined as the peak value of the critical compaction function $\mathcal{C}_{c}(r_m)$ (where $r_m$ is the scale at which the peak occurs), does not necessarily asymptotically saturate to its maximum possible value in the type-I region for sufficiently sharp profiles. Instead, the threshold is found in the type-II region with $\mathcal{C}_{c}(r_m)$ being a minimum. We find, for the cases tested, that this is a general trend associated with profiles that exhibit extremely large curvatures in the linear component of the compaction function $\mathcal{C}_{l}(r) \equiv -4r ζ'(r)/3$ shape around its peak $r_m$ (spiky shapes). To measure this curvature at $r_m$, we define a dimensionless parameter, $κ\equiv -r^{2}_m \mathcal{C}_l''(r_m)$, and we find that the thresholds observed in the type-II region occur for sufficiently large $κ$ for the profiles we have used, contrary to expectations. By defining the threshold in terms of $\mathcal{C}_{l,c}(r_m)$, we extend previous analytical estimations to the type-II region, which is shown to be accurate within a few percent when compared to the numerical simulations for the tested profiles. Our results suggest that current PBH abundance calculations for models where the threshold lies in the type-II region may have been overestimated due to the general assumption that it should saturate at the boundary between the type-I and type-II regions.

Threshold for PBH formation in the type-II region and its analytical estimation

Abstract

We numerically simulate the formation of Primordial Black Holes (PBHs) in a radiation-dominated Universe under the assumption of spherical symmetry, driven by the collapse of adiabatic fluctuations, for different curvature profiles . Our results show that the threshold for PBH formation, defined as the peak value of the critical compaction function (where is the scale at which the peak occurs), does not necessarily asymptotically saturate to its maximum possible value in the type-I region for sufficiently sharp profiles. Instead, the threshold is found in the type-II region with being a minimum. We find, for the cases tested, that this is a general trend associated with profiles that exhibit extremely large curvatures in the linear component of the compaction function shape around its peak (spiky shapes). To measure this curvature at , we define a dimensionless parameter, , and we find that the thresholds observed in the type-II region occur for sufficiently large for the profiles we have used, contrary to expectations. By defining the threshold in terms of , we extend previous analytical estimations to the type-II region, which is shown to be accurate within a few percent when compared to the numerical simulations for the tested profiles. Our results suggest that current PBH abundance calculations for models where the threshold lies in the type-II region may have been overestimated due to the general assumption that it should saturate at the boundary between the type-I and type-II regions.

Paper Structure

This paper contains 9 sections, 15 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Cosmological set up for PBH formation
  3. Curvature profiles and new parameterization
  4. Numerical results
  5. Analytical extension to the type-II region
  6. Conclusions and discussion
  7. Convergence of the numerical simulations
  8. Skeleton for the threshold analytical determination
  9. Curvature profiles parameterized in terms of $\mathcal{C}_{l}(r_m)$ and $\kappa$

Figures (4)

  • Figure 1: Some profiles for the critical compaction function $\mathcal{C}(r)$ (first-panel), the linear component $\mathcal{C}_l(r)$ (second-panel), the $\zeta(r)$ profiles (third-panel) and the density contrast $\delta \rho/\rho_b(r)$ extrapolated at horizon crossing (fourth-panel)
  • Figure 2: Top-panel: Threshold in terms of $\mathcal{C}_{c}$, the bottom and top horizontal dotted lines correspond to $2/5$ and $2/3$, respectively, and the vertical dashed line is located at $\kappa = 30$. Bottom-panel: Threshold in terms of the linear component $\mathcal{C}_{l,c}$, the bottom and top horizontal dotted lines correspond to $(4/3)(1-\sqrt{2/5})$ and $4/3$ respectively, whereas the horizontal dashed line coincides with the top dotted one. The blue solid line in both panels corresponds to the analytical estimate of Eq.\ref{['eq:delta_c']}. The dashed blue line corresponds to the numerical fitting of Eq.\ref{['eq:extension_analytical_estimate']}.
  • Figure 3: Top panel: Analytical estimation Eqs.\ref{['eq:analytical_estimate_deltac']} and \ref{['eq:extension_analytical_estimate']}. Bottom panel: Relative deviation in percentage between the analytical estimation and the numerical results.
  • Figure 4: Top panel: Numerical evolution of the Hamiltonian constraint for different cases using the exponential profiles Eq.\ref{['eq:profile_zeta_exp']} with collapse (solid-lines) and sub-collapse cases (dashed-lines). The $t_H \equiv \left(\epsilon\, e^{\zeta(r_m)}\right)^{-2}$ corresponds to the time of horizon crossing when the fluctuation reenters the cosmological horizon in a homogeneous background and $H_0$ is the initial Hubble parameter (see MIO). The simulations were initialized with parameters $t_0 = 1, a_0 = 1$. Bottom panel: Values of $\Delta \mathcal{C}_{l,c}$ as a function of $\epsilon$.