The trichotomy of primordial black holes initial conditions

TL;DR

The work introduces a trichotomy of primordial black hole initial conditions based on the three-dimensional curvature of a near-super-horizon core: Type-C (closed), Type-O (open), and Type-F (flat). Using gradient expansion and numerical simulations, it shows that PBH formation thresholds depend on both the core and the surrounding shell, with Type-C cores lowering the threshold and Type-O/Type-F generally increasing it, and that the transition between Type-I and Type-II behavior is governed by the core amplitude relative to the shell peak, tracked by the parameter $w_{\cal R}$. The paper layers analytical insights (universal-type thresholds for Type-I via curvature at the compaction maximum) with detailed numerics to map when each core type dominates, revealing a nontrivial transition region and the insufficiency of relying on a single maximum of the compaction function. It discusses statistical implications for different power spectra: narrow spectra favor Type-F as statistically common but non-spherical; very broad spectra, potentially related to NanoGrav signals, can enhance Type-C(I) PBH production and yield an IR-peaked mass spectrum. Overall, the results emphasize that non-linear overdensity and 3D curvature, rather than the compaction function alone, are central to predicting PBH abundances and mass spectra across cosmological scenarios.

Abstract

We show that the threshold to form a black hole, in an asymptotically flat and radiation dominated Friedman-Robertson-Walker (FRW) Universe, is not solely (mainly) determined by the behaviour of the compaction function at its maximum, as earlier thought, but also by the three-dimensional curvature at smaller (but super-horizon) scales, which we call "the core". We find three classes of initial conditions characterized by an open (O), closed (C), or flat (F) FRW core surrounded by a shell with higher three-dimensional curvature. In the C case, the core helps the collapse so that the black hole formation threshold is there the lowest among all cases. Type-II black holes might only be generated by Type-O or F (each of those with different thresholds, with O being the highest) or by a Type-C with an effective F core. Finally, we argue that an F core is typically more probable for a sharp power spectrum, however, it is also more likely related to non-spherical initial conditions. On the other hand, a very broad power spectrum, which might be related to the observed NanoGrav signal, would favor the formation of Type-I black holes with a mass spectrum peaked at the Infra-Red scale.

Figures (12)

• Figure 1: Typical ${\cal R}(r)$ for the three types of perturbation at threshold. All of them correspond to $w\simeq38.6$. The dashed vertical line is $r=r_m$.
• Figure 2: Numerical threshold, in terms of $g$, for the three types of perturbations. The dashed line correspond to $g=\frac{4}{3}.$ All the points correspond to different configurations of the $\alpha$ and $\beta$ parameters (see Appendix \ref{['appendix:parameters']}).
• Figure 3: Numerical threshold values, in terms of $g$, for Type-I perturbations.
• Figure 4: Relative deviation, $d$, for Type-I perturbations, considering the analytical formula \ref{['deltaA']}. The left panel is the relative deviation in terms of $g$ and the right panel in terms of $\mathcal{C}$. Here, one can clearly see that for all profiles considered, the relative deviation $d_\mathcal{C}<2\%$.
• Figure 5: ${\cal R}(r)$ for the three types of perturbation at threshold ($w\simeq38.6$).