A new approach for simulating PBH formation from generic curvature fluctuations with the Misner-Sharp formalism

TL;DR

This work removes a long-standing barrier to simulating type-II curvature fluctuations in the Misner–Sharp formalism by introducing the trace of the extrinsic curvature, $K$, as an auxiliary variable that absorbs divergent terms related to throat structures. The authors implement a pseudospectral code, SPriBHoS-II, and validate it on exponential curvature profiles, enabling stable evolution through horizon formation and allowing a detailed study of type-II PBH thresholds and masses. They identify a beta-dependent transition to the type-II regime around $\beta \approx 2.5$, report type-II cases that do not form PBHs, and find a universal critical scaling for PBH mass near threshold with exponents $\gamma \approx 0.353$–$0.351$, modulated by curvature-profile shape. Overall, the approach broadens the Misner–Sharp toolkit for PBH formation studies and highlights the crucial role of profile shape and pressure gradients in type-II fluctuations, with implications for PBH mass functions and cosmological constraints.

Abstract

Primordial Black Holes (PBHs) may have formed in the early Universe due to the collapse of super-horizon curvature fluctuations. Simulations of PBH formation have been essential for inferring the initial conditions that lead to black hole formation and for studying their properties and impact on our Universe. The Misner-Sharp formalism is commonly used as a standard approach for these simulations. Recently, type-II fluctuations, characterized by a non-monotonic areal radius, have gained interest. In the standard Misner-Sharp approach for simulating PBH formation with these fluctuations, the evolution equations exhibit divergent terms ($0/0$), which complicate and prevent the simulations. We formulate a new approach to overcome this issue in a simple manner by using the trace of the extrinsic curvature as an auxiliary variable, allowing simulations of type-II fluctuations within the Misner-Sharp formalism. Using a set of standard exponential-shaped curvature profiles, we apply and test our new approach and numerical code based on pseudospectral methods to study the time evolution of the gravitational collapse, threshold values of type A/B PBHs and PBH mass. Interestingly, we identify cases of type-II fluctuations that do not necessarily result in PBH formation.

Figures (7)

• Figure 1: Profiles of the curvature $\zeta(r)$ (left-panel) and compaction function (right panel) for different values of $\beta$ and fixing $\mu=1.0$
• Figure 2: Snapshots of the evolution of different quantities for the case $\beta=2$ with $\mu=1.0$ with $r_m = 20 R_{H}(t_0)$, corresponding to a case of PBH formation with type II-A PBH.
• Figure 3: Snapshots of the evolution of different quantities for the case $\beta=1$ with $\mu=2.0$ with $r_m = 10 R_{H}(t_0)$, corresponding to a case of PBH formation with type-II B PBH.
• Figure 4: Snapshots of the evolution of different quantities for the case $\beta=4$ with $\mu=0.45$ with $r_m = 30 R_{H}(t_0)$, corresponding to a case of type-II no PBH.
• Figure 5: Top panels: Snapshots of the Hamiltonian constraint $\mathcal{H}$ at specific times. Bottom panels: Average value of the Hamilton constraint evolution $\mid \mid \mathcal{H}\mid \mid^2$.