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 re-entry 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 anti-correlated 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 $r$, 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

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 $S_{2}$.

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 $S_{2}$.


 : $\nu = 1$

: $\nu = 1$


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

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


 : $\nu = 10$ : 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 section \cref{sec:section3} (solid blue), see in particular \cref{eq:P2_semianalytic}.

: $\nu = 10$ : 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 section \cref{sec:section3} (solid blue), see in particular \cref{eq:P2_semianalytic}.


 : Caption not extracted

: Caption not extracted


Schematic representation of random walks, from $S=0$ to the maximum value $\sigma^2$. Most of the trajectories are confined in the grey 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.

Schematic representation of random walks, from $S=0$ to the maximum value $\sigma^2$. Most of the trajectories are confined in the grey 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.


 : $\lambda = 1$

: $\lambda = 1$


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

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


 : $\lambda = 10$ : 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.

: $\lambda = 10$ : 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.


 : Caption not extracted

: Caption not extracted


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