Harvesting primordial black holes from stochastic trees with FOREST

Author(s)

Animali, Chiara, Auclair, Pierre, Blachier, Baptiste, Vennin, Vincent

Abstract

We introduce a novel framework to implement stochastic inflation on stochastic trees,modelling the inflationary expansion as a branching process. Combined with the δ Nformalism, this allows us to generate real-space maps of the curvature perturbation that fullycapture quantum diffusion and its non-perturbative backreaction during inflation. Unlike latticemethods, trees do not proceed on a fixed background since new spacetime units emerge dynamicallyas trees unfold, naturally incorporating metric fluctuations. The recursive structure ofstochastic trees also offers remarkable numerical efficiency, and we develop the FOrtran RecursiveExploration of Stochastic Trees (FOREST) tool and demonstrate its performance. We showhow primordial black holes blossom at unbalanced nodes of the trees, and how their massdistribution can be obtained while automatically accounting for the “cloud-in-cloud” effect. Inthe “quantum-well” toy model, we find broad mass distributions, with mild power laws terminatedby exponential tails. We finally compare our results with existing approximations in theliterature and discuss several prospects.

Figures

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Elementary vertex of a stochastic tree, where one parent Hubble patch $i$ gives rise to two independent Hubble patches $\ell$ and $m$ after expanding for $\Delta N=\ln(2)/3$.

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Example of a stochastic tree, made of multiple elementary vertices of the type shown in \cref{fig:elementary:tree}. The leaves are Hubble patches where inflation ends, and the patch labels correspond to their topological coordinates in the tree (see main text).

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: $\Delta N$: we divide the $x$-axis.

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: $2 \Delta N$: we divide the $y$-axis.

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: $3 \Delta N$: we divide the $z$-axis.

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: Without PBHs.

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: With PBHs.

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Example of a tree realization with five PBHs in the flat-well toy model, with $\mu=0.8$ and starting from $\phi=\Delta\phi$. This is the same realization as \cref{fig:cmap}. The black patches are those for which the coarse-shelled curvature perturbation, used as a proxy for the compaction function, exceeds the PBH formation threshold.

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: Probability distribution of the final volume.

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: Probability distribution of the volume-averaged expansion.

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Volume-weighted probability distribution of the first-passage time $\mathcal{N}_{x_*\to x_\mathrm{end}}$ through the end-of-inflation hypersurface, in the flat well for $\mu = 0.6, 0.7, 0.8, 0.85, 0.89$ and $x_*=1$. These were obtained with, respectively, $10^{11}, 10^{11}, 5\times 10^{10}, 2\times 10^{10}, 10^{10}$ trees. Full coloured lines stand for the results of \texttt{FOREST}, whereas black dashed lines represent the analytical formula~\eqref{eq:FPTVolWeight}.

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: Mass fraction of PBHs.

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: PBH mass distribution.

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Different prescriptions for the branching time.

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Different prescriptions for the branching time.

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Different prescriptions for the branching time.

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Example tree considered in \cref{sec:discretization:artefact}.

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: $\mu=0.6$, $10^7$ trees

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: $\mu=0.8$, $10^5$ trees

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: $\mu=0.6$, $10^7$ trees

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: $\mu=0.8$, $10^7$ trees

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Demonstration for \cref{eq:alpha} using of the binary representation of nodes $i$ and $j = i+d$. In this example, $i=1101001_2 = 105$ and $j= 1101111_2=111$ are distant by $d=6$. One has $q = 3$ (three green bits) and $q_\mathrm{T} - q = 3$ (three red bits).