Compaction function in stochastic inflation: a \texttt{FOREST} of type I and II primordial black holes

Author(s)

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

Abstract

We show how to compute the compaction function within stochastic inflation, by solving the random field dynamics on stochastic binary trees. In this framework, the compaction function is directly related to the ratio of the volumes emerging from the sibling and child branches of a given node. This construction also determines whether or not the areal radius of a perturbation increases monotonically with the radial coordinate, thereby distinguishing between type-I and type-II fluctuations. As an application, we investigate primordial black hole (PBH) formation in a single-field toy model with a constant potential slope, using stochastic-tree realizations generated with the public code \texttt{FOREST}. In the classical regime, where quantum diffusion is subdominant, the PBH mass function is narrowly distributed and type-II fluctuations are strongly suppressed relative to type I. By contrast, in the quantum and near-critical (i.e. close to eternal inflation) regimes, the PBH mass distribution spans several orders of magnitude, the overall PBH abundance is enhanced, and type-II fluctuations outnumber type I. In that case cloud-in-cloud effects are also important, highlighting the need for a better understanding of the evolution and collapse of type-II fluctuations in order to obtain robust PBH predictions when stochastic effects are significant.

Figures

An example stochastic tree, depicting branching nodes ($i,j,k,\dots$), the \efolds\ between two consecutive bulk nodes ($\Delta N$) and from the last bulk node to a leaf ($\mathcal{N}_{n\to\ell}$), the full set of leaves descending from node $i$ ($\mathcal{L}_i$), the volume emerging from branching nodes ($V_i, V_j, \dots$) and the volume of individual leaves $V(L_\ell)$.
Caption An example stochastic tree, depicting branching nodes ($i,j,k,\dots$), the \efolds\ between two consecutive bulk nodes ($\Delta N$) and from the last bulk node to a leaf ($\mathcal{N}_{n\to\ell}$), the full set of leaves descending from node $i$ ($\mathcal{L}_i$), the volume emerging from branching nodes ($V_i, V_j, \dots$) and the volume of individual leaves $V(L_\ell)$.
The three generations of nodes used in \cref{eq:R_discrete,eq:Cl_discrete}, together with the fiducial mapping of their volumes into real space.
Caption The three generations of nodes used in \cref{eq:R_discrete,eq:Cl_discrete}, together with the fiducial mapping of their volumes into real space.
: Areal radius
Caption : Areal radius
: Compaction
Caption : Compaction
: Linear compaction
Caption : Linear compaction
An example tree arising from the tilted-well potential of \cref{sec:application}. The black and red nodes correspond to type-I and II PBH-forming nodes, respectively. Solid arrows point to left-hand nodes and dashed arrows point to right-hand nodes.
Caption An example tree arising from the tilted-well potential of \cref{sec:application}. The black and red nodes correspond to type-I and II PBH-forming nodes, respectively. Solid arrows point to left-hand nodes and dashed arrows point to right-hand nodes.
Colour plot of \cref{eq:Cl_offbranch} as a function of $V_{b_\mathrm{R}}/V_a$ and $V_{c_\mathrm{R}}/V_a$, where the domain is restricted according to \cref{eq:constraints:volumes} (the grey masked region lies outside that domain). White contour lines indicate constant values of $\mathcal{C}_{\mathrm{l}, b_\mathrm{L}}=0,2/3,4/3$. The colour scale is clipped to the interval $[-5,5]$ for visualisation purposes, so that the uniformly blue and yellow regions correspond to $\mathcal{C}_{\mathrm{l}, b_\mathrm{L}}<-5$ and $\mathcal{C}_{\mathrm{l}, b_\mathrm{L}}>5$ respectively. The type-I and type-II PBHs regions are denoted with black and red hatching respectively.
Caption Colour plot of \cref{eq:Cl_offbranch} as a function of $V_{b_\mathrm{R}}/V_a$ and $V_{c_\mathrm{R}}/V_a$, where the domain is restricted according to \cref{eq:constraints:volumes} (the grey masked region lies outside that domain). White contour lines indicate constant values of $\mathcal{C}_{\mathrm{l}, b_\mathrm{L}}=0,2/3,4/3$. The colour scale is clipped to the interval $[-5,5]$ for visualisation purposes, so that the uniformly blue and yellow regions correspond to $\mathcal{C}_{\mathrm{l}, b_\mathrm{L}}<-5$ and $\mathcal{C}_{\mathrm{l}, b_\mathrm{L}}>5$ respectively. The type-I and type-II PBHs regions are denoted with black and red hatching respectively.
We sample a smoothed version of the linear compaction function over a discrete set of points and we are blind to features at shorter length scales. Here are three possible profiles that provide the same sample of $\mathcal{C}_\mathrm{l}$.
Caption We sample a smoothed version of the linear compaction function over a discrete set of points and we are blind to features at shorter length scales. Here are three possible profiles that provide the same sample of $\mathcal{C}_\mathrm{l}$.
We sample a smoothed version of the linear compaction function over a discrete set of points and we are blind to features at shorter length scales. Here are three possible profiles that provide the same sample of $\mathcal{C}_\mathrm{l}$.
Caption We sample a smoothed version of the linear compaction function over a discrete set of points and we are blind to features at shorter length scales. Here are three possible profiles that provide the same sample of $\mathcal{C}_\mathrm{l}$.
We sample a smoothed version of the linear compaction function over a discrete set of points and we are blind to features at shorter length scales. Here are three possible profiles that provide the same sample of $\mathcal{C}_\mathrm{l}$.
Caption We sample a smoothed version of the linear compaction function over a discrete set of points and we are blind to features at shorter length scales. Here are three possible profiles that provide the same sample of $\mathcal{C}_\mathrm{l}$.
: $d = 0.7$
Caption : $d = 0.7$
: $d = 1.33$
Caption : $d = 1.33$
Mean tree volume in the tilted quantum well, in the $(d, \mu)$ parameter space and starting from the reflective boundary. The region in grey leads to ``eternal inflation'', with $\ev{V} \to \infty$. $\ev{V}$ increases rapidly at the boundary of this region (the ``critical boundary''). At low $\mu$, the system is dominated by quantum diffusion and $\ev{V} \to 1$. If the drift $d$ is non-zero, there exists a value for $\mu$ above which the classical drift dominates and $\ev{V} \approx e^{3 / d}$ as $\mu \to \infty$. The blue line displays $d\mu^2 = 1$. Blue points represent simulations of the flat quantum well ($d=0$) performed in~\cite{Animali:2025pyf}, while red points correspond to the tilted case ($d\neq 0$) simulations presented here and listed in \cref{tab:sims}.
Caption Mean tree volume in the tilted quantum well, in the $(d, \mu)$ parameter space and starting from the reflective boundary. The region in grey leads to ``eternal inflation'', with $\ev{V} \to \infty$. $\ev{V}$ increases rapidly at the boundary of this region (the ``critical boundary''). At low $\mu$, the system is dominated by quantum diffusion and $\ev{V} \to 1$. If the drift $d$ is non-zero, there exists a value for $\mu$ above which the classical drift dominates and $\ev{V} \approx e^{3 / d}$ as $\mu \to \infty$. The blue line displays $d\mu^2 = 1$. Blue points represent simulations of the flat quantum well ($d=0$) performed in~\cite{Animali:2025pyf}, while red points correspond to the tilted case ($d\neq 0$) simulations presented here and listed in \cref{tab:sims}.
: $d = 0.7$
Caption : $d = 0.7$
: $d = 1.33$
Caption : $d = 1.33$
: $d = 0.7$
Caption : $d = 0.7$
: $d = 1.0$
Caption : $d = 1.0$
: $d = 1.33$
Caption : $d = 1.33$
: $d = 2.0$
Caption : $d = 2.0$
: $d = 0.7, \mu=5$
Caption : $d = 0.7, \mu=5$
: $d = 0.7, \mu=10$
Caption : $d = 0.7, \mu=10$
: $d = 1.0, \mu=5$
Caption : $d = 1.0, \mu=5$
: $d = 1.0, \mu=10$
Caption : $d = 1.0, \mu=10$
: $d = 2.0, \mu=5$
Caption : $d = 2.0, \mu=5$
: $d = 2.0, \mu=10$
Caption : $d = 2.0, \mu=10$
Maximally balanced (left) and maximally imbalanced (right) stochastic-tree configurations.
Caption Maximally balanced (left) and maximally imbalanced (right) stochastic-tree configurations.
: $d = 0.7$
Caption : $d = 0.7$
: $d = 1.0$
Caption : $d = 1.0$
: $d = 1.33$
Caption : $d = 1.33$
: $d = 2.0$
Caption : $d = 2.0$
Comparison between the coarse-shelled proxy of ref.~\cite{Animali:2025pyf} (\texttt{FOREST}v1) and the compaction function approach developed in the present work (\texttt{FOREST}v2) on the mass fraction of PBHs, $f_\mathrm{PBH,end}$, for the flat well corresponding to $d=0$.
Caption Comparison between the coarse-shelled proxy of ref.~\cite{Animali:2025pyf} (\texttt{FOREST}v1) and the compaction function approach developed in the present work (\texttt{FOREST}v2) on the mass fraction of PBHs, $f_\mathrm{PBH,end}$, for the flat well corresponding to $d=0$.
: \texttt{FOREST}v1
Caption : \texttt{FOREST}v1
: \texttt{FOREST}v2
Caption : \texttt{FOREST}v2
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