Clustering of primordial black holes: Basic results

TL;DR

This paper addresses whether primordial black holes (PBHs) formed in the radiation era are spatially clustered and how their clustering can be quantified. Using peaks theory bias, it derives the initial PBH two-point function and power spectrum, showing PBH fluctuations are highly biased relative to the radiation field and inherently include an isocurvature component when PBHs constitute dark matter. The analysis decomposes the PBH power spectrum into a Poisson term, a Kaiser-like adiabatic term, and a small-scale contribution, with a horizon-scale cutoff that shapes early-time behavior and isocurvature constraints from the CMB place limits on PBH mass and abundance. The study further discusses the nonlinear evolution of PBH clusters as an N-body problem, predicting enhanced merging and potential gravitational-wave and high-energy signatures, underscoring that PBHs behave differently from standard CDM and that clustering must be accounted for in cosmological constraints.

Abstract

We investigate the spatial clustering properties of primordial black holes (PBHs). With minimal assumptions, we show that PBHs created in the radiation era are highly clustered. Using the peaks theory model of bias, we compute the PBH two-point correlation function and power spectrum. For creation from an initially adiabatic power spectrum of perturbations, the PBH power spectrum contains both isocurvature and adiabatic components. The absence of observed isocurvature fluctuations today constrains the mass range in which PBHs may serve as dark matter. We briefly discuss other consequences of PBH clustering.

Figures (8)

• Figure 1: Allowed region in $\nu - M_{PBH}$ parameter space for PBHs, assuming $f=1$. Solid curve is the upper limit on $\nu$ due to isocurvature perturbations (from Equation (\ref{['fluclimit']})). Other curves are lower limits on $\nu$ due to number density (Equation (\ref{['nstar']})): long dashed line uses the erfc approximation, dotted line uses the BBKS formula with $n=1$, short dashed line uses the GLMS formula with $n=1.5$. Heavy lines show where PBH dark matter is allowed by the isocurvature constraint. Shown also is the temperature of the universe $T$ when PBHs form. The line at $M = M_{*}\sim10^{15}$g is mass below which PBHs would have Hawking evaporated by the current day (assuming no accretion or merging). The line at $M \sim 3 M_\odot$ is the mass above which PBHs would be confused with astrophysical BHs.
• Figure 2: The same as Figure \ref{['fig_nu1']}, except with $f=0.1$.
• Figure 3: The same as Figure \ref{['fig_nu1']}, except with $f=10^{-3.5}$.
• Figure 4: The PBH Power Spectrum for $\nu=1.17$. Dotted line is the radiation power spectrum, consisting of a $n=1$ spectrum with COBE normalization, along with a gaussian spike in power at $k=1$. Solid line is the PBH power spectrum, dashed line is the quadratic estimate of the PBH power spectrum.
• Figure 5: The PBH Power Spectrum for $\nu=2.62$. Dotted line is the radiation power spectrum, consisting of a $n=1$ spectrum with COBE normalization, along with a gaussian spike in power at $k=1$. Solid line is the PBH power spectrum, dashed line is the quadratic estimate of the PBH power spectrum.