Primordial black holes

Author(s)

Dokuchaev, V.I., Eroshenko, Yu.N., Nikulin, V.V., Postnov, K.A., Rubin, S.G., Stasenko, V.D.

Abstract

Primordial black holes, whose formation in the early Universe was first hypothesized by Ya.B.~Zel'dovich and I.D.~Novikov in 1966 (the text of this pioneering article is provided in Appendix II to this review), have attracted increasing attention in recent years. Modern astronomical data across the electromagnetic spectrum, together with observations of gravitational waves and cosmic neutrinos, have revealed a number of unique astrophysical phenomena that may be related to primordial black holes formed in the pre-stellar cosmological epoch. This review describes various models of primordial black hole formation, as well as methods for testing these models with astronomical and astrophysical observations. In particular, we discuss in detail the formation of primordial black holes and their clusters through the collapse of closed domain walls and through effects associated with extra spatial dimensions of the Universe.

Figures

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Constraints on the mass fraction $f_{\rm PBH}$ of PBHs in cold DM as a function of their mass $M_{\rm PBH}$, derived from various effects for a monochromatic PBH mass function. Shown are the constraints from quantum evaporation of PBHs, from gravitational microlensing, from gravitational-wave (GW) detections, and from gas accretion onto PBHs, as well as dynamical constraints. The figure was produced using the publicly available code PBHbounds~\cite{Kav19}.

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The collapsing region is modeled by a part of a closed universe with the scale factor $S(\tau)$. Outside the perturbed region, the Universe is assumed to be flat with the scale factor $R(t)$; transition layers may be present.

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Typical shape of a potential with a local plateau, where, according to \eqref{drhoPhi}, the field fluctuations increase sharply.

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Plots of the potential term and the kinetic factor \eqref{Pot} for the parameters $n=6$, $c_1=-8000$, $c_2=-5000$, $a_2=-500$. Rolling down to the left vacuum $\phi=0$ corresponds to decompactified multidimensional universe; while the right vacuum, $\phi_{\rm min} =2.3\cdot10^{-4}$, corresponds to the observed four-dimensional Universe.

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The self-similar DM density profile around a BH obtained in~\cite{Ber85} is shown by the solid line. The density is given in units of the mean cosmological density. The shell turns around at radius $r=r_s(z)$, i.e., at $r/r_s=1$. The vertical dotted line marks the virial radius $r_{\rm vir}$. To the left of this line is the virialized, phase-mixed region with density (\ref{rhovir}); to the right is the single-stream flow region. The dashed line shows the actual density in the virialized region, which differs from the density given by the self-similar solution.

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Characteristic halo mass as a function of redshift. The solid lines are given by expression~\eqref{eq:M_char} and are labeled with the value of the parameter $M_\pbh f_\pbh/M_{\odot}$. The dotted lines show $N\sigma$ halos from inflationary perturbations, determined by solving the equation $N\sigma_{\rm inf}(M,z) = \delta_c$, where $N = 1,2,3$ is also indicated in the plot.

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Formation of a PBH subsystem at the center of a DM halo due to dynamical friction.

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Initial clustering of PBHs arising in the domain wall mechanism due to the spatial correlation of quantum field fluctuations during inflation. Fluctuations at successive moments during inflation (frames 1--3) push the scalar field over the potential maximum (frame 3). After the end of inflation, the field rolls down to the potential minima, and a cluster of domain walls forms (frame 4), which then collapse into PBHs.

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Initial mass profile $M_{\rm h}(r_i)$ of a PBH cluster (solid curve) and DM mass profile $M_{\rm DM}(r_i)$ (dashed curve).

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Final density profile (\ref{dprofeq}) of the induced halo ($\rho$ in units of $M_{\odot}$~pc$^{-3}$) as a function of the distance $r_c$ from the cluster center for DM (dotted line), for PBHs (dashed line), and for the total density (solid line). The asymptotic power laws are also shown.

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Present-day merger rate of PBHs with mass $M_\pbh = 30 \, M_{\odot}$ as a function of their fraction in DM. The gray region bounded by the dotted lines shows the range of merger rates of late binaries depending on the efficiency of cluster destruction during structure formation; the upper boundary corresponds to the limiting case $w = 1$ (see expression~\eqref{eq:mr_late}), in which cluster destruction is neglected. The horizontal band shows the LIGO/Virgo/KAGRA observations from the O3 run, $ \mathcal{R} = 17.9 -44 $~Gpc$^{-3}$~yr$^{-1}$~\cite{KAGRA:2021duu}. The solid line corresponds to the merger rate of early binaries including the suppression factor $P_{\rm np}$ (mergers of unperturbed binaries); the dash-dotted line, without suppression.

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Redshift dependence of the merger rates of primordial and astrophysical BHs. The gray hatched region shows the merger rate of PBH binaries formed in the early Universe, taking into account their Poisson clustering; the upper boundary corresponds to $f_\pbh = 0.1$, and the lower boundary corresponds to the absence of clustering effects, $f_\pbh < 10^{-3}$. The dashed line shows binaries formed in three-body interactions in a cluster~\eqref{eq:mr3b1}. The blue line shows the scenario with initial PBH clustering for the parameters $\delta_\pbh = 1$, $\sigma$ = 10~km/s, and $M_\pbh = 30M_{\odot}$ (see expressions~\eqref{eq:tau}~and~\eqref{eq:dPdt_Sf}).

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Gravitational-wave background from mergers of PBH binaries with mass $M_\pbh = 30 \, M_{\odot}$ for various clustering scenarios. The gray shaded region corresponds to PBH clustering due to Poisson noise for $f_\pbh = 0.007-0.1$. The blue solid line shows the model with initial PBH clustering with parameters $\delta_\pbh = 1$ and $f_\pbh = 0.1$, where the merger rate is given by the sum of expressions~\eqref{eq:mr_cl} and~\eqref{eq:mr_per_cl}. The red dash-dotted line shows the contribution of astrophysical BHs \eqref{eq:mrabh}, whose merger rate is normalized to $\mathcal{R}_{\rm ABH}(t_0) = 40$~Gpc$^{-3}$~yr$^{-1}$. The dashed lines show the projected sensitivities of some future gravitational-wave detectors; the upper limit from the LIGO/Virgo/KAGRA O4a observing run~\cite{LIGOScientific:2025kry} is also shown.