Black hole formation in the Friedmann universe: Formulation and computation in numerical relativity

TL;DR

This work develops a cosmological numerical-relativity framework based on constant mean curvature slicing to follow primordial black hole formation from well outside the Hubble horizon to horizon crossing and collapse. It provides analytic solutions for nonlinear superhorizon curvature perturbations and a seamless numerical evolution for a radiation-dominated Friedmann background, enabling precise assessment of BH formation thresholds, which depend on the initial density/metric profile. A key finding is that the threshold is not universal but varies with the surrounding density, and a compaction-function criterion offers a robust diagnostic for BH formation in spherical symmetry. Altogether, the results emphasize the importance of spatial correlations in inflationary perturbations for predicting primordial black hole abundances and motivate extensions to non-spherical geometries.

Abstract

We study formation of black holes in the Friedmann universe. We present a formulation of the Einstein equations under the constant mean curvature time-slicing condition. Our formalism not only gives us the analytic solution of the perturbation equations for non-linear density and metric fluctuations on superhorizon scales, but also allows us to carry out a numerical relativity simulation for black hole formation after the scale of the density fluctuations is well within the Hubble horizon scale. We perform a numerical simulation of spherically symmetric black hole formation in the radiation-dominated, spatially flat background universe for a realistic initial condition supplied from the analytic solution. It is found that the initial metric perturbation has to be non-linear (the maximum value of 3D conformal factor $ψ_0$ at $t=0$ should be larger than $\sim 1.4$) for a black hole to be formed, but the threshold amplitude for black hole formation and the final black hole mass considerably depend on the initial density (or metric) profile of the perturbation: The threshold value of $ψ_0$ at $t=0$ for formation of a black hole is smaller for a high density peak surrounded by a low density region than for that surrounded by the average density region of the flat universe. This suggests that it is necessary to take into account the spatial correlation of density fluctuations in the study of primordial black hole formation.

Figures (8)

• Figure 1: The spectrum shapes of the density fluctuation ($\exp(-k^2/4)-\exp(-k^2\sigma^2/4)$) are shown for several $\sigma$. The solid line denotes the case for $\sigma=\infty$ in which the density peak is surrounded by a flat universe, and the dotted lines denote the cases for $1.5 \leq \sigma \leq 8$.
• Figure 2: $\psi_0$ at $t=0$ as a function of $C_{\delta}$ for $\sigma=\infty$, 2, 3 and 5.
• Figure 3: $\delta$ and $2(1-\alpha)$ at origin as a function of time $t$ for black hole formation case ($C_{\delta}=15$, dotted lines) and no formation case ($C_{\delta}=13$, solid lines), respectively.
• Figure 4: The same as Fig. 3, but for $\psi_0$ as a function of time $t$.
• Figure 5: $M_{\rm AH}/r_0$ as a function of time $t/r_0$ in black hole formation cases for $\sigma=\infty$(a), 2(b), 3(c), and 5(d), respectively.