Small-scale clustering of primordial black holes: Cloud-in-cloud and exclusion effects

Author(s)

Auclair, Pierre, Blachier, Baptiste

Abstract

Using an excursion-set approach, we revisit the initial spatial clustering of Primordial Black Holes (PBHs) originating from the Hubble reentry of large Gaussian density fluctuations in the early Universe. We derive the two-point correlation functions of PBHs, properly accounting for the “cloud-in-cloud” mechanism. Our expressions naturally and intrinsically correlate the formation of pairs of PBHs, which is a key difference with the Poisson model of clustering. Our approach effectively includes short-range exclusion effects and clarifies the clustering behaviors at small scale: PBHs are anticorrelated at short distances. Using a scale-independent collapse threshold, we derive explicit expressions for the excess probability to find pairs of PBHs separated by a distance <math display="inline"><mi>r</mi></math>, as well as the excess probability to find pairs with asymmetric mass ratio. Our framework is model independent by construction, and we discuss possible other applications.

Figures

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Schematic representation of random walks performed by smoothed overdensities sharing a common past until ``time'' $S_{r}$ and which subsequently evolve in distinct ways, resulting in two different collapses that occur respectively at the first-crossing times $S_{1}$ and $S_{2}$.

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: $\nu = 1$

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: $\nu = 2$ : $\nu = 5$

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: $\nu = 10$ : Two-point correlation function of the PBHs spatial distribution with constant density threshold $\delta_c$, using the Press-Schechter solution of Ref.~\cite{Ali-Haimoud:2018dau} (dashed red) and the Excursion-Set formalism of \cref{sec:section3} (solid blue), see in particular \cref{eq:P2_semianalytic}.

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Schematic representation of random walks, from $S=0$ to the maximum value $\sigma^2$. Most of the trajectories are confined in the gray region which ranges between $\pm \sigma$. The horizontal lines correspond to different values of $\nu = \delta_c / \sigma$, where $\nu$ is a measure of the barrier's height in units of the typical spread of the trajectories. Low values of $\nu$ indicate that barrier crossings are frequent: hence multiple crossings are also frequent. Higher values of $\nu$ indicate that barrier crossings (and therefore PBH collapses) are rare events, and multiple crossings are suppressed.

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: $\lambda = 1$

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: $\lambda = 2$ : $\lambda = 5$

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: $\lambda = 10$ : Excess probability to find pairs of PBHs with masses $S_1, S_2$ at a fixed distance $r$. $w_n = S_r / S_n$ is a measure of the PBH mass, the limit $w_n \to 0$ corresponds to a very small PBH, whereas $w_n \to 1$ corresponds to the maximum allowed PBH mass in the volume contained within the two PBHs.

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