Numerical simulation of type II primordial black hole formation

TL;DR

The paper investigates PBH formation in a radiation-dominated universe from extreme curvature fluctuations, focusing on type II fluctuations with neck-like areal-radius profiles. It analyzes horizon dynamics using Misner-Sharp mass and null-expansion criteria to classify PBHs into type A (no bifurcating horizon) and type B (with bifurcating horizons), revealing gaps between initial fluctuation type and horizon-formation type in the radiation case. Two initial data profiles are explored: a Gaussian fluctuation and a three-zone-model-inspired profile, showing that PBH mass can both increase and decrease with amplitude depending on the profile, not solely on fluctuation type, and that accretion can be described by $M(t) = 1/(1/M_f + 3F/(2t))$ with $F$ decreasing as amplitude grows. These findings refine the understanding of horizon structures during PBH formation and stress the importance of profile and equation-of-state effects for PBH mass and abundance predictions in the early universe.

Abstract

This study investigates the formation of type II primordial black holes (PBHs) resulting from extremely large amplitudes of initial fluctuations in a radiation-dominated universe. We find that, for a sufficiently large initial amplitude, the configuration of trapping horizons shows characteristic structure due to the existence of bifurcating trapping horizons. We call this type of configuration of the trapping horizons type II-B PBH, while the structure without a bifurcating trapping horizon type II-A PBH. In Ref. [1], in the dust-dominated universe, the type B PBH can be realized by the type II initial fluctuation, which is characterized by a non-monotonic areal radius as a function of the radial coordinate (throat structure) in contrast with the standard case, type A PBH with a monotonic areal radius (type I fluctuation). Our research reveals that a type II fluctuation does not necessarily result in a type B PBH in the radiation-dominated case. We also find that for an initial amplitude well above the threshold value, the resulting PBH mass may either increase or decrease with the initial amplitude, depending on its specific profile rather than its fluctuation type.

Figures (36)

• Figure 1: In the left panel, the vertical and horizontal axes are the functional forms for the curvature perturbation $\zeta(r)$ given by Eq. \ref{['eq:Gprofile']} and the scale-up coordinate $z/L$ defined in Eq. \ref{['eq:scale-up_coordinate']}, respectively. Note that $z$ is a monotonically increasing function of $r$ and that $L$ is chosen for the computation region's length. We plot $\zeta(r)$ for the set of the amplitude parameter values $\mu=0.5, 1.2, 1.4, 1.6$ and $1.8$. In the right panel, the vertical and horizontal axes are the areal radius $R(r)$ and the scale-up coordinate $z/L$, respectively, for the initial data sets of the long-wavelength solutions generated by $\zeta(r)$ plotted in the left panel. We can see that $R(r)$ is a monotonically increasing function for $\mu=0.5$ and $1.2$, while it increases, decreases, and increases again as $r$ increases for $\mu=1.4, 1.6$, and $1.8$. In the latter cases, $R(r)$ has a maximum and a minimum with $\partial_{r}R=0$, where the minimum corresponds to the neck.
• Figure 2: In the left panel, the vertical and horizontal axes are the compaction function $\mathcal{C}_\text{SS}(r)$ and the scale-up coordinate $z/L$, respectively, for the initial data sets of the long-wavelength solutions generated by Eq. \ref{['eq:Gprofile']} for the same set of the amplitude parameter $\mu$. We can see that $\mathcal{C}_\text{SS}(r)$ takes only one maximum, which is smaller than $1/2$, for $\mu=0.5$ and $1.2$, while it takes two distinct maxima, which are equal to $1/2$, and a single minimum between the two maxima for $\tilde{\mu}=1.4, 1.6$ and $1.8$. The latter cases correspond to type II fluctuations as shown in Appendix \ref{['sec:twoPeaks']}. In the right panel, the vertical and horizontal axes are the compaction function $\mathcal{C}_\text{SS}(r)$ and the areal radius $R(r)$, respectively, for the same initial data sets. We can see that $\mathcal{C}_\text{SS}(r)$ takes only the maximum or the two maxima at the larger areal radius $R$ for the larger values of $\mu$. This implies that the mass of the formed black hole increases as $\mu$ increases, as seen in Fig. \ref{['fig:mass_amp']}.
• Figure 3: $\mu=0.5$
• Figure 4: $\mu=1.2$
• Figure 5: $\mu=1.8$